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_images/e3a06fa40efcf6e7a3f62a3e2e24dd25279f6da1db1ed735c287a7e94fa82a53.png rename to _images/fe4f05fe2b33accf2e411ba6247288aacce0f06c210767be95fdae6cef1a2898.png index c9a56db..1522558 100644 Binary files a/_images/e3a06fa40efcf6e7a3f62a3e2e24dd25279f6da1db1ed735c287a7e94fa82a53.png and b/_images/fe4f05fe2b33accf2e411ba6247288aacce0f06c210767be95fdae6cef1a2898.png differ diff --git a/blog/basic-python.html b/blog/basic-python.html index dbe8f20..eb5aafb 100644 --- a/blog/basic-python.html +++ b/blog/basic-python.html @@ -915,7 +915,7 @@

Defining functions in python -../_images/be5189d9ea9e420e6e307b605c5abfc17d94cc94a4840138184a93cb3c6a5a82.png +../_images/c82ad5f4f149f722010224c9bf22c74e45e0d161ead809156d5dd67914ef1e54.png @@ -1042,7 +1042,7 @@

Advanced function creation -../_images/da56408cdd111fb9cd0c7a825af977190e4be0664334b4e8e4a6dfae1a1e1b50.png +../_images/fb1c5ed136fd0b48780878510d4bb46bbb9ebaedc063915bffd511339908a6ed.png
@@ -1056,7 +1056,7 @@

Advanced function creation -../_images/15325932dc2e1d2b3d9607fa216f2fd844f043ec008f5547ad236ec5fb19ea0f.png +../_images/70735392f4187d29834bd045176d16c6e8097afeb2d2c4215a05c10869caa48e.png

In that example we wrap the matplotlib plotting commands in a function, which we can call the way we want to, with arbitrary optional arguments. In this example, you cannot pass keyword arguments that are illegal to the plot command or you will get an error.

@@ -1096,7 +1096,7 @@

Lambda Lambda Lambda
-
<function <lambda> at 0x7f46989c0c20>
+
<function <lambda> at 0x7f36faa9a020>
 4
 

 

@@ -1112,7 +1112,7 @@ 
Lambda Lambda Lambda
-
<function <lambda> at 0x7f469ac29bc0>
+
<function <lambda> at 0x7f36fab7df80>
 5
 

 

@@ -1129,7 +1129,7 @@ 
Lambda Lambda Lambda
-
<function <lambda> at 0x7f46989c0d60>
+
<function <lambda> at 0x7f36faa9a160>
 5
 5
 

@@ -1150,7 +1150,7 @@ 
Lambda Lambda Lambda
 

 

-
<function <lambda> at 0x7f46989c1080>
+
<function <lambda> at 0x7f36faa9a480>
 1
 3
 6
@@ -1326,7 +1326,7 @@ 
Creating arrays in python
/tmp/ipykernel_1909/3002503601.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
+
/tmp/ipykernel_1922/3002503601.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
   print(np.row_stack([a, b]))
 

 

@@ -1509,7 +1509,7 @@ 
Functions on arrays of values
-../_images/0dde0c9d9606e3f45603dff776772f72953e1a858d79bc9be55a4c662ea26006.png
+../_images/8a9d660ddaaa4737db9d130b3f6931574bd728be7e0250294cef3233e9954a07.png
 

 

 
This figure illustrates graphically what the numbers above show. The function crosses zero at approximately \(x = 1.5\). To get a more precise value, we must actually solve the function numerically. We use the function func:scipy.optimize.fsolve to do that. More precisely, we want to solve the equation \(f(x) = \cos(x) = 0\). We create a function that defines that equation, and then use func:scipy.optimize.fsolve to solve it.

@@ -1779,7 +1779,7 @@ 
Indexing vectors and arrays in Python
-1.7950016288086892
 

 

-
/tmp/ipykernel_1909/518394352.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_1922/518394352.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   print(np.trapz( x[x > 2], y[x > 2]))
 

 

@@ -1876,17 +1876,17 @@ 
3D arrays#
 

 

-
[[[0.75563895 0.18498477 0.77047058]
-  [0.26274517 0.89107749 0.34583296]
-  [0.84546459 0.65605378 0.48609064]]
+
[[[0.2227254  0.19726277 0.99153215]
+  [0.97311734 0.38142852 0.56638625]
+  [0.48335471 0.51800664 0.87188098]]
 
- [[0.61448601 0.16467032 0.45140894]
-  [0.50526787 0.78154638 0.79511797]
-  [0.61749088 0.41062892 0.5587581 ]]
+ [[0.37090501 0.88329327 0.93826256]
+  [0.25215415 0.65693695 0.03286427]
+  [0.97872232 0.19969579 0.79929687]]
 
- [[0.24758141 0.21205917 0.49050752]
-  [0.27125643 0.22989573 0.68153965]
-  [0.35608903 0.97072653 0.08109636]]]
+ [[0.69161839 0.70311344 0.87964521]
+  [0.99408152 0.88432325 0.46235968]
+  [0.2289108  0.37483611 0.78532766]]]
 

 

 

@@ -1900,11 +1900,11 @@ 
3D arrays#
 

 

-
[[0.75563895 0.26274517 0.84546459]
- [0.61448601 0.50526787 0.61749088]
- [0.24758141 0.27125643 0.35608903]]
-[0.75563895 0.61448601 0.24758141]
-[0.45140894 0.79511797 0.5587581 ]
+
[[0.2227254  0.97311734 0.48335471]
+ [0.37090501 0.25215415 0.97872232]
+ [0.69161839 0.99408152 0.2289108 ]]
+[0.2227254  0.37090501 0.69161839]
+[0.93826256 0.03286427 0.79929687]
 

 

 

diff --git a/blog/data-analysis.html b/blog/data-analysis.html
index 81ac751..94f584a 100644
--- a/blog/data-analysis.html
+++ b/blog/data-analysis.html
@@ -623,7 +623,7 @@ 
Fit a line to numerical data
-../_images/9cab5f9733126ae4bfbea465ec1662fc1d035ac8145f6083eef298e826e67eb0.png
+../_images/78fdc6defabc7a3280fc782a33aa4feb2e1250a448940c4378e1d478d0b8a436.png
 

 

 
@@ -670,7 +670,7 @@ 
Linear least squares fitting with linear algebra
+../_images/a42c3bb59cc2c093bf6b9be4cef11ca9d5730d46661aaeb77ffebfcf23e02eb1.png
 

 

 
This method can be readily extended to fitting any polynomial model, or other linear model that is fit in a least squares sense. This method does not provide confidence intervals.

@@ -768,7 +768,7 @@ 
Linear regression with confidence intervals.
+../_images/c2fcc135cc0c7607f4a583b24f4e95675d723267656ca42013e030cddad2149a.png
 

 

 
A fourth order polynomial fits the data well, with a good R^2 value. All of the parameters appear to be significant, i.e. zero is not included in any of the parameter confidence intervals. This does not mean this is the best model for the data, just that the model fits well.

@@ -820,7 +820,7 @@ 
Nonlinear curve fitting
Fitted parameters = [-56.46839641   0.57233217   2.7407944   16.55905648]
 

 

-../_images/1f7130eb4df4056f7c7d7e9fa936f170c751bf140ebad1f6007d9d9d56d84e3d.png
+../_images/24eb55b923a124faacb7313ff1c228340e9260844b0fc762c5d67b5eca5d2991.png

See additional examples at \url{http://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html}.

@@ -879,7 +879,7 @@

Nonlinear curve fitting by direct least squares minimization +../_images/777ca1e2dd8aa60b9cbd6adcc4f1e992a8b64aafe6bac269136764841f1821ad.png

@@ -901,7 +901,7 @@

Parameter estimation by directly minimizing summed squared errors -../_images/cdf5aefcbe99ce74ac9a7a9ad2146a8107f711df445488fb7e05df0a1b5b0fc2.png +../_images/0b31ff53625e2d9582b199e5b7ea5280e0d0c3233b3e7bb91948276e0b03b691.png

We are going to fit the function \(y = x^a\) to the data. The best \(a\) will minimize the summed squared error between the model and the fit.

@@ -923,7 +923,7 @@

Parameter estimation by directly minimizing summed squared errors -../_images/2eacd9e5b13ba5d4c83a9be7e22292d0bf5268feb1c2f4812fd8fa67ce5b0926.png +../_images/89fc0f5aacff6472047e29b90385979d36f6e56768e97daf355aa71e8e9f55ef.png

Based on the graph above, you can see a minimum in the summed squared error near \(a = 2.1\). We use that as our initial guess. Since we know the answer is bounded, we use scipy.optimize.fminbound

@@ -948,7 +948,7 @@

Parameter estimation by directly minimizing summed squared errors
2.0900483893314967
-../_images/0a437456b591008802a35ce5c58c38960f2b08801526795aa2eb42409647ea31.png +../_images/e647cd165d4193d18c6e3a656f25867f94b4262f084d94bb3e3d16f15bf6776a.png

We can do nonlinear fitting by directly minimizing the summed squared error between a model and data. This method lacks some of the features of other methods, notably the simple ability to get the confidence interval. However, this method is flexible and may offer more insight into how the solution depends on the parameters.

@@ -1008,7 +1008,7 @@

Nonlinear curve fitting with parameter confidence intervals +../_images/b9035cad5be313768c8516100330a097323a9eb6c9cebf7b21e48c3500c8ec02.png

You can see by inspection that the fit looks pretty reasonable. The parameter confidence intervals are not too big, so we can be pretty confident of their values.

@@ -1064,7 +1064,7 @@

Nonlinear curve fitting with confidence intervals -../_images/7ec782d86cff42f160b56f649ca49452b54f1b071db22ed7312387d577b87a71.png +../_images/ef4bf682b11c7286704710cf714511aee9e12334706dbdc859bb7bcfaf90f535.png @@ -1096,7 +1096,7 @@

Graphical methods to help get initial guesses for multivariate nonlinear reg
-../_images/c6861fa3f7459ae33c59dab3aafe595e094c4f942ce4a7647cf631f31b750208.png +../_images/1dc50d5bb826b2e7e0fd005cacbb350fbfc0e298482bf7f8d8797c8ab36ec47c.png
@@ -1145,7 +1145,7 @@

Graphical methods to help get initial guesses for multivariate nonlinear reg [ 3.25873623 6.59792994 10.29473657 13.68011436 17.29161001 23.62366445]

-../_images/077ee0939432871bbb318adfe1462104f425ee53e0b7b38b8bb7690fc9000804.png +../_images/3790e086efb1285778e0430533e059d8ebb8798e2c0dc89dddcf6c775f0bf48b.png
@@ -1162,7 +1162,7 @@

Graphical methods to help get initial guesses for multivariate nonlinear reg

-../_images/c4ed3188d43409539713e645edfbe69bdc859fdeb19bd0adeb9143df9edbc28d.png +../_images/9cc4b46e8f2dd5906719d58de06694e45b428ae1749071a49a6ab648c4987615.png

It can be difficult to figure out initial guesses for nonlinear fitting problems. For one and two dimensional systems, graphical techniques may be useful to visualize how the summed squared error between the model and data depends on the parameters.

@@ -1207,7 +1207,7 @@

Fitting a numerical ODE solution to data
[2.58893455]
-../_images/4275c4cc292ecf807ce6a72b6dcf8830cc1bc6883cc6e00508fb902357e9d349.png +../_images/755d5e8dd5e949067895d2f138c4378855a60bac487e7702a6afd5f3c75c4546.png diff --git a/blog/differential-equations.html b/blog/differential-equations.html index 0cf3594..70e5ed3 100644 --- a/blog/differential-equations.html +++ b/blog/differential-equations.html @@ -646,7 +646,7 @@

Numerical solution to a simple ode -../_images/195a6e5e52a09792a2642740f4b38cf9fbe6131e9e78314d7a057b63901b2ed3.png +../_images/dd60f24552d73dd2a9499dabb77d010caae47fb887aed03c02ce39d8afaf075e.png

The numerical and analytical solutions agree.

@@ -719,7 +719,7 @@

Plotting ODE solutions in cylindrical coordinates
-../_images/462898867f59524b2d7e4d8902aa74090e87fe284f55fdd83c2ef727c0d8577f.png +../_images/7c8fdb6365cfeca1b64fd9f30b21f174c9fd474504c03679b0817164a4b2fe75.png
@@ -767,7 +767,7 @@

ODEs with discontinuous forcing functions -../_images/5b1237e4255050781c782c671bec3a9229483d04887a6ad825a467bfd49a8ad0.png +../_images/d7665bee0ea373ef1d95f82c1d8dc0b5b0549eaa2fec0853ed9e55e8e7b2abb9.png

You can see the discontinuity in the salt concentration at 10 minutes due to the discontinous change in the entering salt concentration.

@@ -825,7 +825,7 @@

Simulating the events feature of Matlab’s ode solvers -../_images/7f050606f0486595ed84d4ff2df3d741892d5adff046f31ce57028c9d2f52f3d.png +../_images/f63673b0513dd3183ed0cae896746d9ff17c3ff768e007842c4affa59195d64f.png

This particular solution works for this example, probably because it is well behaved. It is “downhill” to the desired solution. It is not obvious this would work for every example, and it is certainly possible the algorithm could go “backward” in time. A better approach might be to integrate forward until you detect a sign change in your event function, and then refine it in a separate loop.

@@ -933,7 +933,7 @@

Mimicking ode events in python +../_images/321a466087ea964d44900d6a58999ff7a55230e9ed3d2b54bea39bc20fe1b8ab.png

That was a lot of programming to do something like find the roots of the function! Today I would use solve_ivp for this.

@@ -970,7 +970,7 @@

Solving an ode for a specific solution value -../_images/6b41634dbebae6440d98892001f43e0db0ffbb487b26118cdb607ca26b5a0019.png +../_images/a54099b19397a7acde81189b36708cff7cb008861f93f73d36b887745e780054.png

You can see the solution is near two seconds. Now we create an interpolating function to evaluate the solution. We will plot the interpolating function on a finer grid to make sure it seems reasonable.

@@ -991,7 +991,7 @@

Solving an ode for a specific solution value -../_images/b476fc1e47f6ef2fddb5fb75e3a2781561230480915e586c1892fc73c5c3e472.png +../_images/ab8c12e9a2931726f98de3a47ad5ad34b0c2e13a53b14328de91b9a2faf2ba44.png

that looks pretty reasonable. Now we solve the problem.

@@ -1159,7 +1159,7 @@

Error tolerance in numerical solutions to ODEs -../_images/a8303cc704d17312fc0e2dd0b0a29fe9b96a3505775807c80e688402cc2bb671.png +../_images/21dc7d50ea671ff369922eaab4f38b8c7db7e9e46225c481371a2ebfc12403b8.png

we want an equation for dPdV, which we will integrate we use symbolic math to do the derivative for us.

@@ -1202,7 +1202,7 @@

Error tolerance in numerical solutions to ODEs -../_images/249f8ff7ac5dda04ecdb1071d73fc13dad71fbf79025f3b46944b1cf2fbe3202.png +../_images/38812c3eee08aa7da7d800df7f5965bed85dd3391d87533f694c746cc3f945f0.png

You can see there is disagreement between the analytical solution and numerical solution. The origin of this problem is accuracy at the initial condition, where the derivative is extremely large.

@@ -1236,7 +1236,7 @@

Error tolerance in numerical solutions to ODEs -../_images/5904ade602c2bebf147116175d594d951a9913c34bb696a7d6ab9093c61da56b.png +../_images/9783100062939cf4fe02e45259930949b6cecb538bc413836a84a33c642e73a1.png

The problem here was the derivative value varied by four orders of magnitude over the integration range, so the default tolerances were insufficient to accurately estimate the numerical derivatives over that range. Tightening the tolerances helped resolve that problem. Another approach might be to split the integration up into different regions. For instance, if instead of starting at Vr = 0.34, which is very close to a sigularity in the van der waal equation at Vr = 1/3, if you start at Vr = 0.5, the solution integrates just fine with the standard tolerances.

@@ -1282,7 +1282,7 @@

Solving parameterized ODEs over and over conveniently -../_images/9a5cf85c2dbfd30d5ad2bc6f3019bcbf15eefc013e32e55d2297026572cf7ccf.png +../_images/9217ff43923323438effc31cbd70ebe650b28cbb61fb25a2a099e8096e3acba4.png

You can see there are some variations in the concentration at t = 0.5. You could over or underestimate the concentration if you have the wrong estimate of \(k\)! You have to use some judgement here to decide how long to run the reaction to ensure a target goal is met.

@@ -1319,7 +1319,7 @@

Yet another way to parameterize an ODE -../_images/5367b0ab34d03645998a02b5acde950a8d8de865e295d7da9f1018cbdcdc5558.png +../_images/2a81d346c950ced9233289603a584dbae54e5e17051f00c162f168c079ed36da.png

I do not think this is a very elegant way to pass parameters around compared to the previous methods, but it nicely illustrates that there is more than one way to do it. And who knows, maybe it will be useful in some other context one day!

@@ -1358,7 +1358,7 @@

Another way to parameterize an ODE - nested function
-../_images/cc064548de6e207fdc4f2ee4e915042b9f01922c7c751a7b1afd19399b868acc.png +../_images/3c0ebcb43be377b144cd0b988685e0799431af6fbcbc6147492aa11ad1c3122b.png

You can see the solution changes dramatically for different values of mu. The point here is not to understand why, but to show an easy way to study a parameterize ode with a nested function. Nested functions can be a great way to “share” variables between functions especially for ODE solving, and nonlinear algebra solving, or any other application where you need a lot of parameters defined in one function in another function.

@@ -1412,8 +1412,8 @@

Solving a second order ode -../_images/6712c6af5471b97d8ce6d6d4400b1fa747717f715f55828736b767fd8a086cf1.png -../_images/266eb8cde8eab3c23eab543f2abdd6ebe663cb1eef7750d87917eee2c5804fa3.png +../_images/2b88fe0be2da8b07b9b2739bff55753c7517f9e74f15aa924f403bfcb0a69fd5.png +../_images/d61b2ea1eb83d98b433b9658d8f0e8c06d467b05db37e69edff37c4a78f875f0.png

Here is the phase portrait. You can see that a limit cycle is approached, indicating periodicity in the solution.

@@ -1460,7 +1460,7 @@

Solving Bessel’s Equation numerically -../_images/bde4821389aa62f785acb3074cd3a8069de6fc9fbe156b5f2694d9d94b6cb931.png +../_images/7cdeb63b86bec3dd31a62cad17fbc4a8e9462141985d7db6cb7bc50bf94d41b3.png

You can see the numerical and analytical solutions overlap, indicating they are at least visually the same.

@@ -1515,7 +1515,7 @@

Phase portraits of a system of ODEs -../_images/289235b666a609dc81efd812f9b8086b8c99d2530bf79b4cb31c4ea13d4687a3.png +../_images/0d0f3983c707d82286951624759b1f301df702b63170fbc78512cf3358d6f047.png

Let us plot a few solutions on the vector field. We will consider the solutions where y1(0)=0, and values of y2(0) = [0 0.5 1 1.5 2 2.5], in otherwords we start the pendulum at an angle of zero, with some angular velocity.

@@ -1538,7 +1538,7 @@

Phase portraits of a system of ODEs -../_images/935d778816de53be1949e6581bb705888a14561d88f12da4015dad7c5dcac07c.png +../_images/aadebc893a2f8f03d9fb81aeaa4c0f7d63574936f50a0b1ab71fb488cfd30f5c.png

What do these figures mean? For starting points near the origin, and small velocities, the pendulum goes into a stable limit cycle. For others, the trajectory appears to fly off into y1 space. Recall that y1 is an angle that has values from \(-\pi\) to \(\pi\). The y1 data in this case is not wrapped around to be in this range.

@@ -1618,11 +1618,11 @@

Linear algebra approaches to solving systems of constant coefficient ODEs
-
/tmp/ipykernel_1964/2289625004.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
+
/tmp/ipykernel_1978/2289625004.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
   T = np.row_stack([t, t])
 

 

-../_images/e5a8938c6cbe07ad66b7a0b98fcd17dd8427b494bd8fa67ca3427ac99dd19eb8.png
+../_images/44792136cd3d48dcc2105d2e7bc26c4c93f9772950db76c1293f23a46b215314.png
@@ -1686,7 +1686,7 @@

First guess -../_images/9efbd2dd7c9d9e88b72dd6c5d17ef420956807cc703938b4119858b874ea69f5.png +../_images/e5f721b2b99ce2696dc4a5b196869e7d96e779ea90839ee2d2cc24dcc6183154.png

Here we have undershot the boundary condition. Let us try a larger guess.

@@ -1710,7 +1710,7 @@

Second guess -../_images/4f1e2a0d640347bdc350d2a72579355f6d8268098c70943b8732c9f24b83f1c3.png +../_images/36849934c84b1c23b5908b9c90250c87a26c0a695847fb95d2e3bc722d730913.png

Now we have clearly overshot. Let us now make a function that will iterate for us to find the right value.

@@ -1923,7 +1923,7 @@

Plane poiseuelle flow solved by finite difference
-../_images/5b4d23f17e3e7d5c40b2545137a01f77725d69f7de0b6227d4d4995b4a9daf87.png +../_images/79b744b52fbc7ab8ba97455dbf93310ee096f7fc45474fea3c0513c21a4df940.png

You can see excellent agreement here between the numerical and analytical solution.

@@ -2002,7 +2002,7 @@

Boundary value problem in heat conduction -../_images/5b424b1e0807a66cbc7f9a17fc19330a65a389b1941b7c73445661cac0c6db0e.png +../_images/563c14e34dd46cdb70e677bd32a8ae84841944df69d35e4a97fd445ce554e109.png @@ -2065,7 +2065,7 @@

A nonlinear BVP -../_images/08a68f79f10b186610b6dd4bb80dd2f4824ff3407b04910f20b59980930af54d.png +../_images/eb98fcb45983b162b49b6f4b2b7fde45547e117f264bccb07ec0bcc52d6bc2a9.png @@ -2134,7 +2134,7 @@

Modeling a transient plug flow reactor -../_images/2e17e38010053a051929d1ad71e85f847606a3a160101a01a2a1139fab16bc48.png +../_images/b8634a196c01065ab7f45592dd4304c6d7b3b27be1625d178bac40989ea1fd15.png

After approximately one space time, the steady state solution is reached at the exit. For completeness, we also examine the steady state solution.

@@ -2150,7 +2150,7 @@

Modeling a transient plug flow reactor -../_images/3e6a2f6cb1c3d68856e173a0b0f20c9fc64569010d2ef58677b512d022ea1e41.png +../_images/23838b95b3ed4ec247e46108fb1b1fb154cc3bb4bb2652169db96dc024384361.png

There is some minor disagreement between the final transient solution and the steady state solution. That is due to the approximation in discretizing the reactor volume. In this example we used 100 nodes. You get better agreement with a larger number of nodes, say 200 or more. Of course, it takes slightly longer to compute then, since the number of coupled odes is equal to the number of nodes.

@@ -2184,10 +2184,10 @@

Modeling a transient plug flow reactor
---------------------------------------------------------------------------
 KeyError                                  Traceback (most recent call last)
-File /opt/hostedtoolcache/Python/3.11.9/x64/lib/python3.11/site-packages/PIL/Image.py:2436, in Image.save(self, fp, format, **params)
-   2435 try:
--> 2436     format = EXTENSION[ext]
-   2437 except KeyError as e:
+File /opt/hostedtoolcache/Python/3.11.9/x64/lib/python3.11/site-packages/PIL/Image.py:2543, in Image.save(self, fp, format, **params)
+   2542 try:
+-> 2543     format = EXTENSION[ext]
+   2544 except KeyError as e:
 
 KeyError: '.mp4'
 
@@ -2227,17 +2227,17 @@ 
Modeling a transient plug flow reactor    516         self.outfile, save_all=True, append_images=self._frames[1:],
     517         duration=int(1000 / self.fps), loop=0)
 
-File /opt/hostedtoolcache/Python/3.11.9/x64/lib/python3.11/site-packages/PIL/Image.py:2439, in Image.save(self, fp, format, **params)
-   2437     except KeyError as e:
-   2438         msg = f"unknown file extension: {ext}"
--> 2439         raise ValueError(msg) from e
-   2441 if format.upper() not in SAVE:
-   2442     init()
+File /opt/hostedtoolcache/Python/3.11.9/x64/lib/python3.11/site-packages/PIL/Image.py:2546, in Image.save(self, fp, format, **params)
+   2544     except KeyError as e:
+   2545         msg = f"unknown file extension: {ext}"
+-> 2546         raise ValueError(msg) from e
+   2548 if format.upper() not in SAVE:
+   2549     init()
 
 ValueError: unknown file extension: .mp4
-../_images/8b538c820219e1f7183502ffe886a86a332ae0796550b1a25e391e4723df0da9.png +../_images/ae85c7ae3c389c403168b2442270f11d2f3178762f45d68db57f5a9d5b96e01f.png

You can see from the animation that after about 10 time units, the solution is not changing further, suggesting steady state has been reached.

@@ -2298,7 +2298,7 @@

Transient heat conduction - partial differential equations -../_images/e2b2cfb2c352a663138aa73fd2ccd812cffd4336931b1ee82fa1e7620630a3c7.png +../_images/de3755f0e821b4538da69b9d00f4712d800baf577bd61fe3155d39ff49b31b42.png
@@ -2317,7 +2317,7 @@

Transient heat conduction - partial differential equations -../_images/31939c54e4aeb8d0a77000dad07b5b0752fe96a47d4eda260bf83436d8c21b37.png +../_images/297b2ccdbda44c4275632cf3c2955c6f1ba2682ed4ac6b46df3749a3c6f52999.png

This version of the graphical solution is not that easy to read, although with some study you can see the solution evolves from the initial condition which is flat, to the steady state solution which is a linear temperature ramp.

@@ -2400,8 +2400,8 @@

Transient diffusion - partial differential equations
0.36136136136136143
-../_images/ef59f651cb0eae73ed5919808b7c4e8477b888a90d57baebaf9dce266e18ada1.png -../_images/b26728f8e11cfbfa1d9964d5141cd8555846f5a0315868cda3d11887c40eaa23.png +../_images/0dfc379c23cbf2dc3100aa61a5ccc547850295a34c11df3bc0096c56a5272131.png +../_images/641d2f7c1c70f8632e26b8fcb494ddcdd236a5748e79e5eb15697ef97a24739d.png

The solution is somewhat sensitive to the choices of time step and spatial discretization. If you make the time step too big, the method is not stable, and large oscillations may occur.

diff --git a/blog/interpolation.html b/blog/interpolation.html index e503eb5..a64adc6 100644 --- a/blog/interpolation.html +++ b/blog/interpolation.html @@ -599,7 +599,7 @@

Better interpolate than never -../_images/b8a07d8cbc6c41df28c6b13937edd2e3565ba605f07cd242764fb97c55ec0693.png +../_images/813863d132bad4c44c2f7d8d0400153a37686c4c15f913c70aa953963ce6805f.png
@@ -671,7 +671,7 @@

Improved interpolation?
-../_images/b9713113f9851e616b28338b9f29de8bd96b5c769808b49b9bd474ac16e1efe6.png +../_images/392d24c6d3af673f332456b82fef2112640fea2a63e467ac4af0a1f906b6ae9a.png

Wow. That is a weird looking fit. Very different from what Matlab produces. This is a good teaching moment not to rely blindly on interpolation! We will rely on the linear interpolation from here out which behaves predictably.

@@ -808,7 +808,7 @@

Discussion -../_images/aa0e1bab164f3265270308f63c0f9f78a7f7adbb8a7163ac776a63a3c2c9d08f.png +../_images/d33502f4ed51078ad20f3cea7f658f908e4e64ff571d0eb193a9ad4e69b30f86.png

You can see that the 1/interpolated f(x) underestimates the value, while interpolated (1/f(x)) overestimates the value. This is an example of where you clearly need more data in that range to make good estimates. Neither interpolation method is doing a great job. The trouble in reality is that you often do not know the real function to do this analysis. Here you can say the time is probably between 3.6 and 5.5 where 1/f(t) = 100, but you can not read much more than that into it. If you need a more precise answer, you need better data, or you need to use an approach other than interpolation. For example, you could fit an exponential function to the data and use that to estimate values at other times.

@@ -916,13 +916,13 @@

Interpolation with splines
/tmp/ipykernel_1994/1138919460.py:26: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+
/tmp/ipykernel_2009/1138919460.py:26: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   plt.title('Max point = ({0:1.2f}, {1:1.2f})'.format(float(xmax),
-/tmp/ipykernel_1994/1138919460.py:27: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+/tmp/ipykernel_2009/1138919460.py:27: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   float(f(xmax))));
 

 

-../_images/ff97e36ccc721f4109b9ad5b09bb061b8c367475182f8ffebda2aad238903f6f.png
+../_images/47ad1dd28fda131569b8aea1f426e674a2393677f9db32e1782807a505f1296d.png

There are other good examples at http://docs.scipy.org/doc/scipy/reference/tutorial/interpolate.html

diff --git a/blog/linear-algebra.html b/blog/linear-algebra.html index a3d3946..00d71b7 100644 --- a/blog/linear-algebra.html +++ b/blog/linear-algebra.html @@ -1178,7 +1178,7 @@

Determining linear independence of a set of vectors -
/tmp/ipykernel_2020/3718588497.py:6: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
+
/tmp/ipykernel_2034/3718588497.py:6: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
   A = np.row_stack([v1, v2, v3])
 

 

@@ -1231,7 +1231,7 @@ 
Another example
[7.57773162 5.99149259]
-
/tmp/ipykernel_2020/73961882.py:7: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
+
/tmp/ipykernel_2034/73961882.py:7: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
   A = np.row_stack([v1, v2])
 

 

diff --git a/blog/math.html b/blog/math.html
index 627c8a7..69b0349 100644
--- a/blog/math.html
+++ b/blog/math.html
@@ -673,12 +673,12 @@ 
Numeric derivatives by differences
-
 Forward difference took 0.000202 seconds
- Backward difference took 0.000180 seconds
- Centered difference took 0.000225 seconds
+
 Forward difference took 0.000212 seconds
+ Backward difference took 0.000192 seconds
+ Centered difference took 0.000232 seconds
 

 

-../_images/82cc01e9a70af72babb19d11f2f1a5bc180f8073d593930b8a0310a827a72140.png
+../_images/e82a955efa2e01fb55f09524ce9405030516d0ae15614c934dfaf2d14380e228.png

@@ -722,7 +722,7 @@

Vectorized numeric derivatives -../_images/48472db659fa43f0cb5f7c93f96bd01509fe7362b7afc510c7c575fd79d6e914.png +../_images/ecf8f289b085fe846d589ffb52a11298e04bcd87b3c9ac6c2b456b3707475af2.png @@ -775,7 +775,7 @@

2-point vs. 4-point numerical derivatives -../_images/b094d29bc2ec3a715c7dacbd9829755e9af7fb7c937d30cf0eddb0850b026e1a.png +../_images/8951267ae3a4c990233213f6c1aaec4fe0074d0c4bff4d030c72b2f0fcf46d14.png @@ -813,8 +813,8 @@

Derivatives by polynomial fitting -../_images/393ae5088971869e22e97945c822de88dc1b0d0ae7893e5bd302d575e55ea115.png -../_images/82a82fe0ac10c27504deab2e738707bf4400a172fe2ab156d88c8c3a0c134f7e.png +../_images/7738a0cfc980d171207cd6a901fb63569c4662bc255cbe3600550ae1c81e2017.png +../_images/ea47e6bdc20b74f0409972c6d997b1a49f952232e08e1de6adf21896a679cc53.png

You can see a third order polynomial is a reasonable fit here. There are only 6 data points here, so any higher order risks overfitting. Here is the comparison of the numerical derivative and the fitted derivative. We have “resampled” the fitted derivative to show the actual shape. Note the derivative appears to go through a maximum near t = 0.9. In this case, that is probably unphysical as the data is related to the consumption of species A in a reaction. The derivative should increase monotonically to zero. The increase is an artefact of the fitting process. End points are especially sensitive to this kind of error.

@@ -855,8 +855,8 @@

Derivatives by fitting a function and taking the analytical derivative
-../_images/c0a2b4d47063a3fce9925756d9b2c5f04bfc7a62ebd8dbe9b2f761881ea0692b.png -../_images/6a144522c7dc5e645b8f12f37db90284780b9d7ff5ec23d02128aaf175acc559.png +../_images/da26d735bbe85b233704b22b589ed320ee30d87041e0b6968233b381de09615e.png +../_images/66504d6b93e26b5fc809cf53489029b8149a999dd3795157d08dee0d829d6556.png

Visually this fit is about the same as a third order polynomial. Note the difference in the derivative though. We can readily extrapolate this derivative and get reasonable predictions of the derivative. That is true in this case because we fitted a physically relevant model for concentration vs. time for an irreversible, first order reaction.

@@ -900,7 +900,7 @@

Derivatives by FFT -../_images/8e6faa1b0d64ec9230f595d385fb7d0e2a941de2bb485f820097ce7640ea3b13.png +../_images/705ba698a2e01fc56d4be41638b5b18fe6685ca4a9f0b8cd8101b48ec1ad490b.png @@ -1011,7 +1011,7 @@

Vectorized piecewise functions -../_images/ac8bffe721f87dbb0d6dc1c557a6fd880497330cb08eb11239ed43ae4ee6b5b9.png +../_images/fc9ceefc99e979f8295d2dbbd46babe82367154dcf2c13dfbd097d77f82b2d1d.png

Neither of those methods is convenient. It would be nicer if the function was vectorized, which would allow the direct notation f1([0, 1, 2, 3, 4]). A simple way to achieve this is through the use of logical arrays. We create logical arrays from comparison statements.

@@ -1035,7 +1035,7 @@

Vectorized piecewise functions
[0. 0. 1. 0. 0. 0.]
-../_images/ac8bffe721f87dbb0d6dc1c557a6fd880497330cb08eb11239ed43ae4ee6b5b9.png +../_images/fc9ceefc99e979f8295d2dbbd46babe82367154dcf2c13dfbd097d77f82b2d1d.png

A third approach is to use Heaviside functions. The Heaviside function is defined to be zero for x less than some value, and 0.5 for x=0, and 1 for x >= 0. If you can live with y=0.5 for x=0, you can define a vectorized function in terms of Heaviside functions like this.

@@ -1068,6 +1068,9 @@

Vectorized piecewise functions
(1.0, 1.1102230246251565e-14)
+

+
+
@@ -1077,7 +1080,7 @@

Vectorized piecewise functions -../_images/ac8bffe721f87dbb0d6dc1c557a6fd880497330cb08eb11239ed43ae4ee6b5b9.png +../_images/fc9ceefc99e979f8295d2dbbd46babe82367154dcf2c13dfbd097d77f82b2d1d.png

There are many ways to define piecewise functions, and vectorization is not always necessary. The advantages of vectorization are usually notational simplicity and speed; loops in python are usually very slow compared to vectorized functions.

@@ -1132,7 +1135,7 @@

Smooth transitions between discontinuous functions
-../_images/cdc15e8419b0808f983d04f10a8954743470dd513b1bb51fedd6b41be986bd9c.png +../_images/bcc8fec0f7f15e94698f3de911996a52882947a424bfd446ba02c59658ecd323.png

You can see the discontinuity at Re = 3000. What we need is a method to join these two functions smoothly. We can do that with a sigmoid function. @@ -1150,7 +1153,7 @@

Smooth transitions between discontinuous functions
-../_images/7c201e653799328fc322e49c2cf179118c45c90f9caf99b7fe41206267c20834.png +../_images/e99cd3898c1d9947dacc058d8d81091a36eeda206f2eb76b4fe7d1562ca2a0a0.png

If we have two functions, \(f_1(x)\) and \(f_2(x)\) we want to smoothly join, we do it like this: \(f(x) = (1-\sigma(x))f_1(x) + \sigma(x)f_2(x)\). There is no formal justification for this form of joining, it is simply a mathematical convenience to get a numerically smooth function. Other functions besides the sigmoid function could also be used, as long as they smoothly transition from 0 to 1, or from 1 to zero.

@@ -1177,7 +1180,7 @@

Smooth transitions between discontinuous functions
-../_images/87c41172c53dfe8a0a068b29ae7076ce79bfcda5f68660ddd0a6e14a90dde2c7.png +../_images/06e47b6a6a2858e1c0aa86c15e61bd8607ef3e9c057cf8bb3628af747e791e44.png

You can see that away from the transition the combined function is practically equivalent to the original two functions. That is because away from the transition the sigmoid function is 0 or 1. Near Re = 3000 is a smooth transition from one curve to the other curve.

@@ -1213,7 +1216,7 @@

Smooth transitions between two constants -../_images/12c5b363fd979ae7e1c078ebcb4727de70aef729b95d7c17923e16ba995ba003.png +../_images/c9c6afdc95ab3dd9bba1e90638bcab12b7548c044a99a3b38a38de762d312dbe.png

This is a nice trick to get an analytical function with continuous derivatives for a transition between two constants. You could have the transition occur at a value other than D = 1, as well by changing the argument to the exponential function.

@@ -1280,7 +1283,7 @@

Numerical data integration
4.25 0.0625
-
/tmp/ipykernel_2047/1388173814.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2059/1388173814.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   i2 = np.trapz(y, x)
 

 

@@ -1304,7 +1307,7 @@ 
Numerical data integration
-../_images/1dc996a31ab3a3fef2a3bdc71195624297ae4b8d64e7dc23cb53be68e7f1944b.png
+../_images/c69a755855b6ed6a1995c2d70bca31e035a7266e7b26122c9ebca7e4e25f06eb.png

The trapezoid method is overestimating the area significantly. With more points, we get much closer to the analytical value.

@@ -1323,7 +1326,7 @@

Numerical data integration
4.000408121620243
-
/tmp/ipykernel_2047/1343167527.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2059/1343167527.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   print(np.trapz(y2, x2))
 

 

@@ -1363,7 +1366,7 @@ 
Combining numerical data with quad
/tmp/ipykernel_2047/1681875887.py:13: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2059/1681875887.py:13: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   print(np.trapz(yfine, xfine))
 

 

@@ -1515,7 +1518,7 @@ 
Wilkinson’s polynomial
 

 

-../_images/339b14aa8a9aada245a175e437482dbdfef9a0257f744b117864d34b93c9ad6a.png
+../_images/82462f493900bf50a0c5a128e5419808d5d4a4a78b235262bafb1db5053ae085.png

Let us consider the expanded version of the polynomial. We will use sympy to expand the polynomial.

@@ -1774,7 +1777,7 @@

The trapezoidal method of integration -
time elapsed = 0.001287698745727539 sec
+
time elapsed = 0.0006933212280273438 sec
 1.9999983517708524
 

 

@@ -1795,7 +1798,7 @@ 
The trapezoidal method of integration
-
time elapsed = 0.00020503997802734375 sec
+
time elapsed = 0.00019860267639160156 sec
 1.9999934070923728
 

 

@@ -1814,7 +1817,7 @@ 
The trapezoidal method of integration
-
time elapsed = 8.702278137207031e-05 sec
+
time elapsed = 7.152557373046875e-05 sec
 1.999993407092373
 

 

@@ -2009,7 +2012,7 @@ 
Function integration by the Romberg method
/tmp/ipykernel_2047/1760157840.py:9: DeprecationWarning: `scipy.integrate.romberg` is deprecated as of SciPy 1.12.0and will be removed in SciPy 1.15.0. Please use`scipy.integrate.quad` instead.
+
/tmp/ipykernel_2059/1760157840.py:9: DeprecationWarning: `scipy.integrate.romberg` is deprecated as of SciPy 1.12.0and will be removed in SciPy 1.15.0. Please use`scipy.integrate.quad` instead.
   print(romberg(np.sin, a, b))
 

 

diff --git a/blog/nonlinear-algebra.html b/blog/nonlinear-algebra.html
index 96100cc..bb006cd 100644
--- a/blog/nonlinear-algebra.html
+++ b/blog/nonlinear-algebra.html
@@ -627,7 +627,7 @@ 
Know your tolerance
-../_images/8f9a9c1e15a1ce8d9abfd0ec30a8532ca5c0ed1e2e3249775f7cebf0fd0c925f.png
+../_images/596dba2b8711799fa117189a3cc3f228609116b748af213b88ceb109d8675607.png
 

 

 
Now let us solve the equation. It looks like an answer is near C=500.

@@ -694,7 +694,7 @@ 
Solving integral equations with fsolve
-../_images/d2e2897fbbea33df12eb15e609acbe2772e88e3a60c4520cf569ab9270d342ea.png
+../_images/31cff26510496e76298f84e833a5a8f754987c1737d8ef5accb38244bc596812.png
 

 

 
Now we can see a zero is near Fa = 0.1, so we proceed to solve the equation.

@@ -765,15 +765,15 @@ 
Method of continuity for nonlinear equation solving
 

 

-
/tmp/ipykernel_2075/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2087/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
   improvement from the last ten iterations.
   y1, = fsolve(tmp, x0)
-/tmp/ipykernel_2075/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
+/tmp/ipykernel_2087/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
   improvement from the last five Jacobian evaluations.
   y1, = fsolve(tmp, x0)
 

 

-../_images/9f8776f7746ce48f45d51931427459ff4b70559ff193e096f0ecf22e5e66c2ad.png
+../_images/95143c46ce94e94f492c7c991441b48a12f418cc757d37b0d6a925a345b0fcc8.png
 

 

 
You can see there is a solution near x = -1, y = 0, because both functions equal zero there. We can even use that guess with fsolve. It is disappointly easy! But, keep in mind that in 3 or more dimensions, you cannot perform this visualization, and another method could be required.

@@ -898,7 +898,7 @@ 
Method of continuity for solving nonlinear equations - Part II
-../_images/792d046195c465d0f95a887c20217d942478cca72f138d5cab35436a4a99b04e.png
+../_images/4e7d168c56cac7c9476226a24893a7636d355ae3714af432253f28af7ffda5fd.png

We found one solution at x=2. What about the other solution? To get that we have to introduce \(\lambda\) into the equations in another way. We could try: \(f(x;\lambda) = x^2 + \lambda(-5x + 6)\), but this leads to an ODE that is singular at the initial starting point. Another approach is \(f(x;\lambda) = x^2 + 6 + \lambda(-5x)\), but now the solution at \(\lambda=0\) is imaginary, and we do not have a way to integrate that! What we can do instead is add and subtract a number like this: \(f(x;\lambda) = x^2 - 4 + \lambda(-5x + 6 + 4)\). Now at \(\lambda=0\), we have a simple equation with roots at \(\pm 2\), and we already know that \(x=2\) is a solution. So, we create our ODE on \(dx/d\lambda\) with initial condition \(x(0) = -2\).

@@ -922,7 +922,7 @@

Method of continuity for solving nonlinear equations - Part II -../_images/ac17de93ba27ca505d3ea643a7f16737e63dc1d2892d46ac76a96adac5c31d55.png +../_images/d1f050a304b6a610155e5a0413a1c7a9032ef96c268073a32beb044925e576af.png

Now we have the other solution. Note if you choose the other root, \(x=2\), you find that 2 is a root, and learn nothing new. You could choose other values to add, e.g., if you chose to add and subtract 16, then you would find that one starting point leads to one root, and the other starting point leads to the other root. This method does not solve all problems associated with nonlinear root solving, namely, how many roots are there, and which one is “best” or physically reasonable? But it does give a way to solve an equation where you have no idea what an initial guess should be. You can see, however, that just like you can get different answers from different initial guesses, here you can get different answers by setting up the equations differently.

@@ -962,7 +962,7 @@

Use roots for this polynomial -../_images/db191f5a48a7313fd34ffc32d0c84cad613dcd7c4441d3f6659122d311cf6cda.png +../_images/cb24aaead3f49f15b3988742bf411552a85b49eb463cafb01ebcf67167b6624e.png

Now we consider several approaches to counting the number of roots in this interval. Visually it is pretty easy, you just look for where the function crosses zero. Computationally, it is tricker.

@@ -1046,7 +1046,7 @@

Finding the nth root of a periodic function -../_images/696f5aa256798d0001502403deab69d09c4d226f3381e60974d5f22547bf6056.png +../_images/7593b21e108f9e170dfbfbb64766e5fe86afd3a8b5a62e16bf2e1190cb5cc552.png

You can see there are many roots to this equation, and we want to be sure we get the nth root. This function is pretty well behaved, so if you make a good guess about the solution you will get an answer, but if you make a bad guess, you may get the wrong root. We examine next a way to do it without guessing the solution. What we want is the solution to \(f(x) = 0\), but we want all the solutions in a given interval. We derive a new equation, \(f'(x) = 0\), with initial condition \(f(0) = f0\), and integrate the ODE with an event function that identifies all zeros of \(f\) for us. The derivative of our function is \(df/dx = d/dx(x J_1(x)) - Bi J'_0(x)\). It is known (http://www.markrobrien.com/besselfunct.pdf) that \(d/dx(x J_1(x)) = x J_0(x)\), and \(J'_0(x) = -J_1(x)\). All we have to do now is set up the problem and run it.

@@ -1086,7 +1086,7 @@

Finding the nth root of a periodic function -../_images/e74f4f3d1e378b5eb2b7dfb354f16c7dde8695b716532de80873ca9044fbc5a6.png +../_images/7e64bf53667d43e52a048719d67ed4b13e0d77573e83f1b983e1552a3326c7c7.png

You can work this out once, and then you have all the roots in the interval and you can select the one you want.

diff --git a/blog/optimization.html b/blog/optimization.html index 4dcbf1e..b08a3ec 100644 --- a/blog/optimization.html +++ b/blog/optimization.html @@ -632,7 +632,7 @@

Finding the maximum power of a photovoltaic device.
-../_images/5f9a720bcdee69ee52e66ee1498e22ceed5f97b4b19adb911eeb5b6d891f51e8.png +../_images/3f173b35ccfa395bf337982f2023cdd742214cff9c567dfe02886cb2230e3432.png

Now, let us be sure there is a maximum in power.

@@ -651,7 +651,7 @@

Finding the maximum power of a photovoltaic device.
-../_images/f9a979e24de48cf0952970d64c40fa08ae4f70142e0653f69dd5093c00458b4f.png +../_images/434ce4ed3c12c6408d7c8b7ca208baca4f7f60f8029f4e5d9b1526a5fbebed59.png

You can see in fact there is a maximum, near V=0.6. We could solve this problem analytically by taking the appropriate derivative and solving it for zero. That still might require solving a nonlinear problem though. We will directly setup and solve the constrained optimization.

@@ -693,7 +693,7 @@

Finding the maximum power of a photovoltaic device. -../_images/ce05a1314f25c9c3e9d370693012af3d7a5ac2d96f0e6cc747c4dbd352761686.png +../_images/dee2faf1f80d40a8967cc93c695b21fbf3332ca552f58b5a855ddd75436f509d.png

You can see the maximum power is approximately 0.2 (unspecified units), at the conditions indicated by the red dot in the figure above.

@@ -728,7 +728,7 @@

Using Lagrange multipliers in optimization -../_images/b9f319ae89cf16e418046504af6bf55485ac9ce744f35ce6c9197bc4f1f20f1b.png +../_images/d9aae9b7fb515c19b0ca9340abc5bbede5a263e4aa5bce7becdba563a2b0f936.png
@@ -916,7 +916,7 @@

Find the minimum distance from a point to a curve. -../_images/38537ec80b3631ed77ef67e96b16ca51e0b9eb612664413aa04732757e1d82c2.png +../_images/5ae0c09600fd410ea23e354da445976e6a472e4ff5fc0c10c8c6f92101d14a44.png

In the code above, we demonstrate that the point we find on the curve that minimizes the distance satisfies the property that a vector from that point to our other point is normal to the tangent of the curve at that point. This is shown by the fact that the dot product of the two vectors is very close to zero. It is not zero because of the accuracy criteria that is used to stop the minimization is not high enough.

diff --git a/blog/plotting.html b/blog/plotting.html index cfffe90..f585e3e 100644 --- a/blog/plotting.html +++ b/blog/plotting.html @@ -591,7 +591,7 @@

Plot customizations - Modifying line, text and figure properties -../_images/4ad376b58ae56511a36324cf1bd04647dfb0f8ec2274a2f70ecaf73731358a00.png +../_images/d53b1d7c51f848fe995e7d81e979681f49308abbd3079ff120dc340651359abb.png

Lets increase the line thickness, change the line color to red, and make the markers red circles with black outlines. I also like figures in presentations to be 6 inches high, and 4 inches wide.

@@ -612,7 +612,7 @@

Plot customizations - Modifying line, text and figure properties -../_images/ee698be0cbdaf99b134c497559acc60f82bd744671875d3e06979ce5e122aa65.png +../_images/869486ddaf8db9d1b354b46e15366ad73435a6f38dd270f8a0117e45970e5eaa.png
@@ -853,7 +853,7 @@

Setting all the text properties in a figure.
findfont: Font family 'Arial' not found.
-../_images/1a3c8761eee8d8d887b913e8f019e927b4533f9aafa273018776ee64b6afab94.png +../_images/57c5ed211821913dc33734b34110b63a8568700ec273db4fd93a57272e627d4e.png

There are many other things you can do!

@@ -879,7 +879,7 @@

Plotting two datasets with very different scales -../_images/9ee66e73babc6e3c8f216d255d28ac86a8e28f7753fb952a81602a20a64a01d1.png +../_images/577acf98068674a51bbd800ed6f531a442fd1467b5c062318d66f81e6ae554fd.png
@@ -898,8 +898,8 @@

Make two plots! -../_images/f4b368f610d88da56059f7878c497e5807292b0b1b613c2730b9b0b9ff2111a1.png -../_images/e9ea264a5a549421977013425fb928b2a4cac00840852fdbfb8c953b6eeaba67.png +../_images/a38a7baa5cd2a6ca2f5cdccbe0a9e862ee56911a2b28d71288894b807b72d7db.png +../_images/a7b0e4fffa599e001171221d8d55b8915d39815d05728d65ff76259750518795.png

@@ -915,7 +915,7 @@

Scaling the results -../_images/80781ebf33466d2f9b3b8ea4613624356798051a698480feb0e36fc549ca58b7.png +../_images/43f2a4fd8e76f4a5500b1df62ad1753fba0f7f69b173161b977ea572ba080900.png

@@ -938,7 +938,7 @@

Double-y axis plot -../_images/9f2fd374102f10d49f21c6be4a9921db99d3b8a8bb02186127416a5357ae0e9f.png +../_images/f0d9a44e6f2ac2616e853c713651d3ca238fa8807ed6dd9e27b7e42f6c4a4f4e.png

@@ -957,7 +957,7 @@

Subplots
-../_images/fb422232c67b75aae6d72f465267f5cec389e8f986e6528a073c099f3a342809.png +../_images/7dc3ce4b1e43ba993585bd5b34d89200a837475c33abb6acbb4727db56c3ab41.png
@@ -1003,7 +1003,7 @@

Customizing plots after the fact -../_images/d3ae5ec1bcae9ce6933f831b86d1a8148b97840ec4b558a5f799fad599d8ebe2.png +../_images/8b4430e725371304b770c9d83f074af6c8873c76da54492f11699f37915b9f98.png @@ -1025,7 +1025,7 @@

Fancy, built-in colors in Python
dict_keys(['aliceblue', 'antiquewhite', 'aqua', 'aquamarine', 'azure', 'beige', 'bisque', 'black', 'blanchedalmond', 'blue', 'blueviolet', 'brown', 'burlywood', 'cadetblue', 'chartreuse', 'chocolate', 'coral', 'cornflowerblue', 'cornsilk', 'crimson', 'cyan', 'darkblue', 'darkcyan', 'darkgoldenrod', 'darkgray', 'darkgreen', 'darkgrey', 'darkkhaki', 'darkmagenta', 'darkolivegreen', 'darkorange', 'darkorchid', 'darkred', 'darksalmon', 'darkseagreen', 'darkslateblue', 'darkslategray', 'darkslategrey', 'darkturquoise', 'darkviolet', 'deeppink', 'deepskyblue', 'dimgray', 'dimgrey', 'dodgerblue', 'firebrick', 'floralwhite', 'forestgreen', 'fuchsia', 'gainsboro', 'ghostwhite', 'gold', 'goldenrod', 'gray', 'green', 'greenyellow', 'grey', 'honeydew', 'hotpink', 'indianred', 'indigo', 'ivory', 'khaki', 'lavender', 'lavenderblush', 'lawngreen', 'lemonchiffon', 'lightblue', 'lightcoral', 'lightcyan', 'lightgoldenrodyellow', 'lightgray', 'lightgreen', 'lightgrey', 'lightpink', 'lightsalmon', 'lightseagreen', 'lightskyblue', 'lightslategray', 'lightslategrey', 'lightsteelblue', 'lightyellow', 'lime', 'limegreen', 'linen', 'magenta', 'maroon', 'mediumaquamarine', 'mediumblue', 'mediumorchid', 'mediumpurple', 'mediumseagreen', 'mediumslateblue', 'mediumspringgreen', 'mediumturquoise', 'mediumvioletred', 'midnightblue', 'mintcream', 'mistyrose', 'moccasin', 'navajowhite', 'navy', 'oldlace', 'olive', 'olivedrab', 'orange', 'orangered', 'orchid', 'palegoldenrod', 'palegreen', 'paleturquoise', 'palevioletred', 'papayawhip', 'peachpuff', 'peru', 'pink', 'plum', 'powderblue', 'purple', 'rebeccapurple', 'red', 'rosybrown', 'royalblue', 'saddlebrown', 'salmon', 'sandybrown', 'seagreen', 'seashell', 'sienna', 'silver', 'skyblue', 'slateblue', 'slategray', 'slategrey', 'snow', 'springgreen', 'steelblue', 'tan', 'teal', 'thistle', 'tomato', 'turquoise', 'violet', 'wheat', 'white', 'whitesmoke', 'yellow', 'yellowgreen'])
-../_images/9e7da72746ccd70dcd052fc134af1a3c3bfa2b049549918fee4766a694fd88f8.png +../_images/8e002cbdbbd8c12c16de21e6e1adaa1111ece0aa8890435cf8e967e5057d0ff0.png @@ -1045,7 +1045,7 @@

Picasso’s short lived blue period with Python -../_images/0d2c1a3f4fea54f0582957ee080b2c8b3dd216d55131e8d699096d9d2f916f31.png +../_images/9f2335f4052a9fc26af4630cd11583c9e52ae4974c4e1bd177ac00c2d857cdf7.png

Picasso copied the table available at http://en.wikipedia.org/wiki/List_of_colors>and parsed it into a dictionary of hex codes for new colors. That allowed him to specify a list of beautiful blues for his graph. Picasso eventually gave up on python as an artform, and moved on to painting.

@@ -1103,7 +1103,7 @@

Picasso’s short lived blue period with Python -../_images/0d2c1a3f4fea54f0582957ee080b2c8b3dd216d55131e8d699096d9d2f916f31.png +../_images/9f2335f4052a9fc26af4630cd11583c9e52ae4974c4e1bd177ac00c2d857cdf7.png @@ -1173,11 +1173,11 @@

Peak annotation in matplotlib -
/tmp/ipykernel_2132/256016704.py:19: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2145/256016704.py:19: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   area = np.trapz(i[ind], w[ind])
 

 

-../_images/59b2238bd6466f1daf9b5c418840f1220bf927e031d379f613031d2bb54d93a7.png
+../_images/4f3db3407e2c9495526935a285836cfe9c309a3171445cdecc37236845b708e5.png
diff --git a/blog/programming.html b/blog/programming.html index c5f5a66..11f725b 100644 --- a/blog/programming.html +++ b/blog/programming.html @@ -795,7 +795,7 @@

Unique entries in a vector -
['d', 'b', 'abracadabra', 'a', 'c']
+
['d', 'a', 'b', 'abracadabra', 'c']
@@ -1142,7 +1142,7 @@

About your python -
3.11.9 (main, Jun 20 2024, 16:02:53) [GCC 11.4.0]
+
3.11.9 (main, Jun 25 2024, 18:25:01) [GCC 11.4.0]
 /opt/hostedtoolcache/Python/3.11.9/x64/bin/python
 linux
 /opt/hostedtoolcache/Python/3.11.9/x64
@@ -1168,16 +1168,16 @@ 
About your python
-
uname_result(system='Linux', node='fv-az1980-869', release='6.5.0-1022-azure', version='#23~22.04.1-Ubuntu SMP Thu May  9 17:59:24 UTC 2024', machine='x86_64')
+
uname_result(system='Linux', node='fv-az887-444', release='6.5.0-1022-azure', version='#23~22.04.1-Ubuntu SMP Thu May  9 17:59:24 UTC 2024', machine='x86_64')
 Linux
 

 

 
('64bit', 'ELF')
 x86_64
-fv-az1980-869
+fv-az887-444
 Linux-6.5.0-1022-azure-x86_64-with-glibc2.35
 x86_64
-('main', 'Jun 20 2024 16:02:53')
+('main', 'Jun 25 2024 18:25:01')
 3.11.9
 

 

@@ -1240,9 +1240,7 @@ 
About your python
 
initially inside /home/runner/work/pycse/pycse/pycse-jb/pycse___python_computations_in_science_and_engineering/blog
 inside /home/runner/work/pycse/pycse/pycse-jb/pycse___python_computations_in_science_and_engineering/blog/data
-

-

-
['PT.txt', 'example4.xls', 'antoine_data.dat', 'example4.xlsx', 'test.docx', 'debug.txt', 'antoine_database.mat', 'testdata.txt', 'gc-data-21.txt', 'debug-3.txt', 'example3.xls', 'debug-2.txt', 'raman.txt', 'example.xlsx', 'debug-4.txt', 'example2.xls']
+['PT.txt', 'example4.xls', 'antoine_data.dat', 'example4.xlsx', 'test.docx', 'debug.txt', 'antoine_database.mat', 'testdata.txt', 'gc-data-21.txt', 'debug-3.txt', 'example3.xls', 'debug-2.txt', 'raman.txt', 'example.xlsx', 'debug-4.txt', 'example2.xls']
 Exception caught:  <class 'Exception'>
 Running final code
 finally inside /home/runner/work/pycse/pycse/pycse-jb/pycse___python_computations_in_science_and_engineering/blog
diff --git a/blog/statistics.html b/blog/statistics.html
index 3e39b95..63e20eb 100644
--- a/blog/statistics.html
+++ b/blog/statistics.html
@@ -860,7 +860,7 @@ 
Model selection
-../_images/9db82f219058c1365621cc94dd720749c031a264fe4be17d9942c48eb624c553.png
+../_images/f577b58eda514cb633f8929e019841b437cadfc88507abbd6f34f276ed622683.png
 

 

 
It appears the data is roughly linear, and we know from the ideal gas law that PV = nRT, or P = nR/V*T, which says P should be linearly correlated with V. Note that the temperature data is in degC, not in K, so it is not expected that P=0 at T = 0. We will use linear algebra to compute the line coefficients.

@@ -931,7 +931,7 @@ 
Model selection
-../_images/93a0fc6cf8a376c530b87fea790aa707468eb65682e43a642982dfa9ea484a02.png
+../_images/593ee4d7807c5852b25695e4e7e2ecd92bd30f1126475b959b66a4c304d88260.png
 

 

 
The fit looks good, and R^2 is near one, but is it a good model? There are a few ways to examine this. We want to make sure that there are no systematic trends in the errors between the fit and the data, and we want to make sure there are not hidden correlations with other variables. The residuals are the error between the fit and the data. The residuals should not show any patterns when plotted against any variables, and they do not in this case.

@@ -957,7 +957,7 @@ 
Model selection
-../_images/d5610b757f93ea82dead3906bc1ced81cfdd0a67f16054ef1b2893ada23aa37c.png
+../_images/ab671f862654c6441b304f6d36444806e6dc751c993b0ec9dd6f16f27c670cc4.png
 

 

 
There may be some correlations in the residuals with the run order. That could indicate an experimental source of error.

@@ -972,7 +972,7 @@ 
Model selection
-../_images/6dff13a1ea9f076d15f5899ee5ce7c7fa5e778e69fb4ceb3d54f60af1fe0c371.png
+../_images/5ba9e2f7e8b6f897fd81930619acd4119e5d3f03dfbe32c705bb275a4ca9f3f7.png

It is hard to argue there is any correlation here.

@@ -1067,7 +1067,7 @@

Model selection
[4.056801244813359 4.123083499127842]

-../_images/c710e89f6e381b21f16a802a8e17f3e7fe07d880764ac704f14c23aa0de4d930.png +../_images/4391bee2b2ed564b2f727ea47d1de7aaf80918651122c68d60869f2bafa61beb.png

The fit is visually still pretty good, and the R^2 value is only slightly worse. Let us examine the residuals again.

@@ -1083,7 +1083,7 @@

Model selection -../_images/44e6f9a0e6d170f4b4a56fc3643554ac3fbb6edcc5643558fbb6a916b378f726.png +../_images/7cb54573db434c88927bb151c7d16bc8d018242fb0d76a8050811c44bb619ae1.png

You can see a slight trend of decreasing value of the residuals as the Temperature increases. This may indicate a deficiency in the model with no intercept. For the ideal gas law in degC: \(PV = nR(T+273)\) or \(P = nR/V*T + 273*nR/V\), so the intercept is expected to be non-zero in this case. Specifically, we expect the intercept to be 273*R*n/V. Since the molar density of a gas is pretty small, the intercept may be close to, but not equal to zero. That is why the fit still looks ok, but is not as good as letting the intercept be a fitting parameter. That is an example of the deficiency in our model.

@@ -1124,9 +1124,9 @@

Addition and subtraction -../_images/694920f52f387d54f1744d81ba8be84f266003694743e77a7f033f9d7598d92a.png -
0.4987406586557553
-0.49774280605070226
+../_images/4b031a057a58aa98b838c722b5a9f13679bb8d84607636b70ca6df5346872cf7.png
+
0.505004004389188
+0.4946534726349484
 0.5
 

 

@@ -1150,7 +1150,7 @@ 
Multiplication
-
11.855132435744245
+
11.943419918779492
 11.872657663724665
 

 

@@ -1169,7 +1169,7 @@ 
Division
 

 

 

-
0.2911705382447411
+
0.29247477698630503
 0.2898598062432779
 

 

@@ -1194,7 +1194,7 @@ 
exponents
-
1.7489408816596042
+
1.7070256434054254
 1.7236544062149992
 

 

@@ -1208,7 +1208,7 @@ 
exponents
-
0.007463235917889261
+
0.007479008114283664
 0.007483647387490024
 

 

@@ -1236,7 +1236,7 @@ 
the chain rule in error propagation
-
3.610284165422935
+
3.6088721879129313
 3.6180105030251086
 

 

@@ -1494,7 +1494,7 @@ 
Another approach to error propagation
-
Ca(exit) = 0.0050070399814704306+/-0.00016843324995414444
+
Ca(exit) = 0.00500533174368263+/-0.00016868059772141597
 

 

 

@@ -1526,7 +1526,7 @@ 
Random thoughts
-
n = 0.36419554585156433
+
n = 0.02772888200446888
 you lose.
 

 

@@ -1551,10 +1551,10 @@ 
Random thoughts
-
You won 5119 times (51.190000%)
+
You won 5100 times (51.000000%)
 

 

-../_images/c975bc369997f82994bb8500510e8acf54f35e4574683f9beeeb7381422f9637.png
+../_images/f59f1438e0d121b3a543a2f4871cb2605a865f38cd42f9625480c98c20ad0ad5.png
 

 

 
As you can see you win slightly more than you lost.

@@ -1570,10 +1570,10 @@ 
Random thoughts
-
71
-[97 28 42]
-[[55  2]
- [ 4 14]]
+
74
+[ 1 39 83]
+[[55 38]
+ [65 74]]
 

 

 

@@ -1591,8 +1591,8 @@ 
Random thoughts
-
1.04053993511862
-[0.90196414 1.36945963]
+
1.7333673534149683
+[0.77883842 0.9988782 ]
 

 

 

@@ -1615,7 +1615,7 @@ 
Random thoughts
-../_images/71dca0bf74a040ea191c69bd61db7a3d661739b6a5d63fb0f2de420003cf5c77.png
+../_images/4beb90caa40af7d423cbb7d1881f20728c53de5c9d81084d8905be265e3ffd51.png
 

 

 
What fraction of points lie between plus and minus one standard deviation of the mean?

@@ -1638,8 +1638,8 @@ 
Random thoughts
-
67.600000% of samples are within +- 1 standard deviations of the mean
-95.400000% of samples are within +- 2 standard deviations of the mean
+
68.720000% of samples are within +- 1 standard deviations of the mean
+95.600000% of samples are within +- 2 standard deviations of the mean
 

 

 

diff --git a/blog/worked-examples.html b/blog/worked-examples.html
index 5ae992e..6f0dde7 100644
--- a/blog/worked-examples.html
+++ b/blog/worked-examples.html
@@ -1161,7 +1161,7 @@ 
Estimating the boiling point of water
-../_images/1f9ffd8e1468c10c19c5bde9f241d79eca8d6d2de4704c950866dfb564b38d85.png
+../_images/63ee541f78f4893b3bdd3b9acf03374622487f7bce222fd415a1886d82395b18.png
 

 

 

@@ -1432,7 +1432,7 @@ 
Linear equality constraints for atomic mass conservation
/tmp/ipykernel_2224/1892264215.py:6: RuntimeWarning: invalid value encountered in log
+
/tmp/ipykernel_2238/1892264215.py:6: RuntimeWarning: invalid value encountered in log
   G = np.sum(nj * (Gjo / R / T + np.log(nj / Enj)))
 

 

@@ -1548,13 +1548,13 @@ 
The Gibbs free energy of a reacting mixture and the equilibrium composition<
 

 

 

-
/tmp/ipykernel_2224/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
+
/tmp/ipykernel_2238/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
   diffg += R * T * x1 * np.log(x2 * P / P0)
-/tmp/ipykernel_2224/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
+/tmp/ipykernel_2238/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
   diffg += R * T * x1 * np.log(x2 * P / P0)
 

 

-../_images/52c598538c6758b04880a8e4def82ea3cec988e62f3245883f7ff10385a3daac.png
+../_images/cd92d89850b4705c8f81544bfaf046cf2c4e8be0d25acb81da9cb8cc90c5ce5f.png
 

 

 
Now we simply minimize our Gwigglewiggle function. Based on the figure above, the miminum is near 0.45.

@@ -1576,13 +1576,13 @@ 
The Gibbs free energy of a reacting mixture and the equilibrium composition<
 
0.46959618248998036
 

 

-
/tmp/ipykernel_2224/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
+
/tmp/ipykernel_2238/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
   diffg += R * T * x1 * np.log(x2 * P / P0)
-/tmp/ipykernel_2224/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
+/tmp/ipykernel_2238/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
   diffg += R * T * x1 * np.log(x2 * P / P0)
 

 

-../_images/5db2d30bf3f7307d4957cf1cfcb4f97c23579cf39772530ecbc2384920a1dd9a.png
+../_images/afe3203baf56e59c2f40feba583c6ff3c319d1444c5f2026c820602ee506f9ce.png
 

 

 
To compute equilibrium mole fractions we do this:

@@ -1809,7 +1809,7 @@ 
Plot how the \(\Delta G\)
 

 

 

-../_images/ccb8c6b9767683f4416378e424e0a84e860389269ba378878312187caf1c99b8.png
+../_images/305a5e91e6d530c7f45c995ea569d84130f37e159f543f5ffca37394fb17d615.png
 

 

 
Over this temperature range the reaction is exothermic, although near 1000K it is just barely exothermic. At higher temperatures we expect the reaction to become endothermic.

@@ -1831,7 +1831,7 @@ 
Equilibrium constant calculation
-../_images/be045851dc6657fb27082aa89ff603204c15ae955cbf8e49e73d025792304ea9.png
+../_images/e360f72a6250eb2f7e2b0176a342a126f4bb9237ee7e7116af54039df4204d09.png
 

 

 
@@ -2019,9 +2019,9 @@ 
Constrained minimization to find equilibrium compositions
/tmp/ipykernel_2224/1928982228.py:17: RuntimeWarning: invalid value encountered in log
+
/tmp/ipykernel_2238/1928982228.py:17: RuntimeWarning: invalid value encountered in log
   d * np.log(P) + yI * d * np.log(yI) +
-/tmp/ipykernel_2224/1928982228.py:18: RuntimeWarning: invalid value encountered in log
+/tmp/ipykernel_2238/1928982228.py:18: RuntimeWarning: invalid value encountered in log
   yB * d * np.log(yB) + yP1 * d * np.log(yP1) + yP2 * d * np.log(yP2))
 

 

@@ -2075,7 +2075,7 @@ 
Constrained minimization to find equilibrium compositions
-../_images/d034062fc47ce81c7a7840434f6490754ed2c93b956dcdf5f10bc1fc9db0eb44.png
+../_images/bd9106e5a131e45bbe5d766009761ee23bade4dfd5dacc72d7fca8d7f470d326.png
 

 

 
You can see we found the minimum. We can compute the mole fractions pretty easily.

@@ -2373,13 +2373,13 @@ 
Numerically calculating an effectiveness factor for a porous catalyst bead

 

-
/tmp/ipykernel_2224/4291268947.py:40: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2238/4291268947.py:40: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   eta_numerical = (np.trapz(k * Ca * 4 * np.pi * (r**2), r)
-/tmp/ipykernel_2224/4291268947.py:41: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+/tmp/ipykernel_2238/4291268947.py:41: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   / np.trapz(k * CAs * 4 * np.pi * (r**2), r))
 

 

-../_images/43af5f6fa255345218f95e9e611c98e9c6fd313c1923020bb3de48a8c9613cb4.png
+../_images/06a8548a1303d2680ff898e1f7b239a6dde58936c02035b017044c216a56afa6.png
 

 

 
You can see the concentration of A inside the particle is significantly lower than outside the particle. That is because it is reacting away faster than it can diffuse into the particle. Hence, the overall reaction rate in the particle is lower than it would be without the diffusion limit.

@@ -2458,7 +2458,7 @@ 
Computing a pipe diameter
The minimum pipe diameter is 0.0389653369530596 m
 

 

-../_images/f4b38e186e6b7dbb5412e30655b455211e1f6f945f330aa7b70d77c913c05527.png
+../_images/fd748d6de2fd99787778056585795d7c8bf213e30008f0e11b9f14f99205436e.png
 

 

 
Any pipe diameter smaller than that value will result in a larger pressure drop at the same volumetric flow rate, or a smaller volumetric flowrate at the same pressure drop. Either way, it will not meet the design specification.

@@ -2779,7 +2779,7 @@ 
The equal area method for the van der Waals equation
-../_images/63e93e6cb70712d42896a2ca4d7913f2441b95423e7f0f09ce4d4e785129abae.png +../_images/e76fbe7f01f28b461e071a741748d6d11349afaed19387dc92906164bd1c59b2.png

The idea is to pick a Pr and draw a line through the EOS. We want the areas between the line and EOS to be equal on each side of the middle intersection. Let us draw a line on the figure at y = 0.65.

@@ -2796,7 +2796,7 @@

The equal area method for the van der Waals equation
-../_images/0072a3c19fa4e21919464ac65f6184c6f95758b7d3fba4a10a50b3178e398c18.png +../_images/605d27d71edd657b69c712e2743e0c45b1be0fbc6e24c00417cfc5a2c0c55eaf.png

To find the areas, we need to know where the intersection of the vdW eqn with the horizontal line. This is the same as asking what are the roots of the vdW equation at that Pr. We need all three intersections so we can integrate from the first root to the middle root, and then the middle root to the third root. We take advantage of the polynomial nature of the vdW equation, which allows us to use the roots command to get all the roots at once. The polynomial is \(V_R^3 - \frac{1}{3}(1+8 T_R/P_R) + 3/P_R - 1/P_R = 0\). We use the coefficients t0 get the roots like this.

@@ -2815,7 +2815,7 @@

The equal area method for the van der Waals equation
[0.60286812 1.09743234 2.32534056]
-../_images/7c7d048cd291065cb68d8a2986cb1d4b7335e212e08042a4008b618a074a33c9.png +../_images/43b0ea57bb809982ef8e2a523fe8160fe99de588d210cd82ca1bba3f908fac09.png
@@ -2978,7 +2978,7 @@

Compute areasNameError: name 'y_eq' is not defined -../_images/8e786b4b1247bf8cfcdfd162cd3087d81628a2419079bab93073ac55a9c20dbe.png +../_images/b6d802a3e849820f5decbab77df186ce091e9c951ea940ed032a3bd827923693.png

@@ -3033,7 +3033,7 @@

Time dependent concentration in a first order reversible reaction in a batch
-../_images/c4de420792db4cab9f3c1121db028a608a14bca712403a6bb939d449ba70f07f.png +../_images/f2087d82cfde5ac2531a39029e133827916bc02ab7fe47d6a58e2c97db541381.png

That is it. The main difference between this and Matlab is the order of arguments in odeint is different, and the ode function has differently ordered arguments.

@@ -3167,7 +3167,7 @@

Integrating the batch reactor mole balance -../_images/1c0707b2558e31df7769dc92bd7090f0d85b8525ec30451b8cdc371db08e7b58.png +../_images/375deb776a401f2ee4164ebe76c38e7b31e52cc12056283eab4260328d82066f.png

You can read off of this figure to find the time required to achieve a particular conversion.

@@ -3214,7 +3214,7 @@

Plug flow reactor with a pressure drop -../_images/9b93f709ece9fc89c4876dab1169c348b86df7a88ffc9dddf64ec7ad3b088da3.png +../_images/4c9ee639930344419fee972673c3a5cb7678beed49858cd2508a098a778cc4fe.png

See Fogler, 4th edition. page 193.

@@ -3585,7 +3585,7 @@

What region is a point in -../_images/b874296dcfcf3785256a1525c0a7128663d0ff321776f111a946b9ffabb51d0a.png +../_images/dc413a1ee7932b817b9d23878470bddd49d9ea3960b19ea45fb3fa884a39e2d3.png

In this example, the boundary is complicated, and not described by a simple function. We will check for intersections of the line from the arbitrary point to the reference point with each segment defining the boundary. If there is an intersection in the boundary, we count that as a crossing. We choose the origin (0, 0) in this case for the reference point. For an arbitrary point (x1, y1), the equation of the line is therefore (provided x1 !=0):

@@ -3698,13 +3698,13 @@

What region is a point in -
/tmp/ipykernel_2224/339476128.py:64: RuntimeWarning: invalid value encountered in scalar divide
+
/tmp/ipykernel_2238/339476128.py:64: RuntimeWarning: invalid value encountered in scalar divide
   m1 = y1 / x1
-/tmp/ipykernel_2224/339476128.py:64: RuntimeWarning: divide by zero encountered in scalar divide
+/tmp/ipykernel_2238/339476128.py:64: RuntimeWarning: divide by zero encountered in scalar divide
   m1 = y1 / x1
 

 

-../_images/5bd899a62d4c3d2323541bd318da0bea180dc24a09371cdb6d90810277c24bf9.png
+../_images/2b2630012b05b400f9a826d57cd1544695e0fb28f2aeedefbcbffe10677d27a4.png

If you look carefully, there are two blue points in the red region, which means there is some edge case we do not capture in our function. Kudos to the person who figures it out. diff --git a/book/00-intro.html b/book/00-intro.html index 40fab1b..2d4bca0 100644 --- a/book/00-intro.html +++ b/book/00-intro.html @@ -761,7 +761,7 @@

plotting
-../_images/3fdbb32ed56648ea66f2e69492efa1b1172105d44b4cd3a4867b5131ab5f4568.png +../_images/b0b62248a05cdf34fba1a7671badfa43a97a7db8c6ca90ab7f261ad409772d80.png
@@ -793,7 +793,7 @@

scipy#<
-../_images/9b9e7c3a02c6eba62efab3888d334e2b435fbdbdab3fd716e9189162b960a71f.png +../_images/da094fc6601412b8e827f00d4a6d4e0a201c9d7ddd748a0ae0a07ef5e7fa7910.png
@@ -807,16 +807,16 @@

scipy#< 
{'neval': 63,
  'last': 2,
- 'iord': array([          1,           2,           0,           0,           4,
-                547,         413,           0,           0,           0,
-                  0,           0,  -509864192,       21887,  -509864192,
-              21887,           0,           0,           0,           0,
-                  4,        2188, -1435875184,       32539,  1594927664,
-              32539,  -510875488,       21887,           0,           0,
-             903644,           0,           1,           0,           0,
-                  0,           0,           0,           0,           0,
-                  0,           0,           0,           0,           0,
-                  0,           0,           0,           0,           0],
+ 'iord': array([          1,           2,           0,           0,   218176832,
+           50334979,     4328707, -1610151165, -2130509563,   168558857,
+          134618112,   269356036,   196103132,   229657428, -1185391788,
+        -1057027072,  1020549432,   288550084,   788541753,  -331557660,
+          809046578,   123229003,   130614288,  1506346256,   708837666,
+            2752523,     5832776,     7798889,   369557642,   637545217,
+          906177281,  1325485569,  1443712780,  1963025921, -2063369727,
+            6098177,   553716061,   856765187,   588325653,    53813521,
+           16973318,  -704760322,  1593690581,  -470014715,   872205571,
+           27328168,       50016,       32738,    85458288,       32738],
        dtype=int32),
  'alist': array([0.00000000e+00, 2.25000000e+00, 7.67216743e-04, 2.10783652e-03,
         4.30872995e-03, 7.48622952e-03, 1.17307690e-02, 1.71108947e-02,
@@ -844,19 +844,19 @@ 
scipy#<
         2.01589600e-312, 2.33419537e-312, 2.22809558e-312, 2.14321575e-312,
         2.41907520e-312, 2.33419537e-312, 9.97338022e-313, 2.35541533e-312,
         2.27053550e-312, 1.64726287e+265]),
- 'rlist': array([2.05718868e-001, 9.12099070e-001, 6.90490532e-310, 6.90490504e-310,
-        6.90490485e-310, 6.90490532e-310, 6.90490504e-310, 6.90490485e-310,
-        6.90490532e-310, 6.90490504e-310, 6.90490485e-310, 6.90490532e-310,
-        6.90490504e-310, 6.90490485e-310, 6.90490532e-310, 6.90490504e-310,
-        6.90490485e-310, 6.90490532e-310, 6.90490504e-310, 6.90490485e-310,
-        6.90490532e-310, 6.90490504e-310, 6.90490485e-310, 6.90490532e-310,
-        6.90490504e-310, 6.90490485e-310, 6.90490532e-310, 6.90490504e-310,
-        6.90490485e-310, 6.90490532e-310, 6.90490504e-310, 6.90490485e-310,
-        6.90490532e-310, 6.90490504e-310, 6.90490485e-310, 6.90490532e-310,
-        6.90490504e-310, 6.90490485e-310, 6.90490532e-310, 6.90490504e-310,
-        6.90490485e-310, 6.90490532e-310, 6.90490504e-310, 6.90490485e-310,
-        6.90490532e-310, 6.90490504e-310, 6.90490485e-310, 6.90490532e-310,
-        6.90490504e-310, 6.90490485e-310]),
+ 'rlist': array([2.05718868e-001, 9.12099070e-001, 6.94705636e-310, 6.94705588e-310,
+        6.94705568e-310, 6.94705636e-310, 6.94705588e-310, 6.94705568e-310,
+        6.94705636e-310, 6.94705588e-310, 6.94705569e-310, 6.94705636e-310,
+        6.94705588e-310, 6.94705569e-310, 6.94705636e-310, 6.94705588e-310,
+        6.94705569e-310, 6.94705636e-310, 6.94705588e-310, 6.94705569e-310,
+        6.94705636e-310, 6.94705588e-310, 6.94705569e-310, 6.94705636e-310,
+        6.94705588e-310, 6.94705569e-310, 6.94705636e-310, 6.94705588e-310,
+        6.94705569e-310, 6.94705636e-310, 6.94705588e-310, 6.94705569e-310,
+        6.94705636e-310, 6.94705588e-310, 6.94705569e-310, 6.94705636e-310,
+        6.94705588e-310, 6.94705569e-310, 6.94705636e-310, 6.94705588e-310,
+        6.94705569e-310, 6.94705636e-310, 6.94705588e-310, 6.94705569e-310,
+        6.94705636e-310, 6.94705588e-310, 6.94705569e-310, 6.94705636e-310,
+        6.94705588e-310, 6.94705569e-310]),
  'elist': array([7.86630709e-09, 1.01263339e-14, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
@@ -882,7 +882,7 @@ 
scipy#<
-
/tmp/ipykernel_2255/843280888.py:2: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2268/843280888.py:2: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   (np.trapz(integrand(x1), x1) - I) / I
 

 

diff --git a/book/01-jupyter.html b/book/01-jupyter.html
index f1d4dfb..ec00d0c 100644
--- a/book/01-jupyter.html
+++ b/book/01-jupyter.html
@@ -644,7 +644,7 @@ 
Markdown
-../_images/d2b6e72a533d9907f8ceab56f2e5c08e8aee6c2f220c517393143eba3038f8a2.png +../_images/12310fabde9b4db49066dcf3b88ab2f61f7194b60765668fb68e2977a7584db3.png
diff --git a/book/02-integration-1.html b/book/02-integration-1.html index 49ed7f9..196f025 100644 --- a/book/02-integration-1.html +++ b/book/02-integration-1.html @@ -609,7 +609,7 @@

Numerical integration of data -../_images/b038f9dd42392e43a167f1cda431fb7b0912119385a7dd90ac50d9ae9b796077.png +../_images/3216836a9a8315734f6982cd3e712774a7d2985a09186949677564718096bdf4.png

If we want the area under this curve, it is represented by:

@@ -708,7 +708,7 @@

Numerical integration of data -../_images/d84f34dda1ac42cafbcdd69aad213de52659ca0a4e9f894a1c8f9037dd4879f5.png +../_images/ea9c2b2f4d32ba1c5775bdedb9ef13aff7c82142347d23457b5c184faa0b4adc.png

Why don’t these agree? The trapezoid method is an approximation of the integral. In this case the straight lines connecting the points overestimate the value of the function, and so the area under this curve is overestimated.

@@ -727,7 +727,7 @@

numpy.trapz -
/tmp/ipykernel_2306/3604360318.py:4: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2315/3604360318.py:4: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   np.trapz(y, x)
 

 

@@ -882,7 +882,7 @@ 
Estimating the volume of a plug flow reactor
(0.0, 2.0)
-../_images/e1b816faa13e9dc74bf3608a482ae5c35bdb9d99809ee388e7697b25e1a6bf74.png +../_images/3c7f1dd11f19209b80d643b1ca4652ff4d36109e9d46c8137878520d0aaccc21.png

We could iterate over the conversions and print the volume for each value. This is a little wasteful since we recompute the areas in each iteration, but here it is so fast it does not matter.

@@ -953,14 +953,14 @@

Estimating the volume of a plug flow reactor
/tmp/ipykernel_2306/4086341036.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2315/4086341036.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   vol = np.trapz(y[0:i+1], X[0:i+1])
 

 

 
Text(0, 0.5, 'V (m^3)')
 

 

-../_images/bcbf14798c63e658a88847e9ab8d92d2ad5d41762ecc733dc575ba85fcf27e72.png
+../_images/7925349caf05552ea1c4fffd358dc8d04fa332fd610c9b57bd10b7df048e8f32.png
@@ -990,14 +990,14 @@

Estimating the volume of a plug flow reactor
/tmp/ipykernel_2306/427682055.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2315/427682055.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   vol = np.trapz(y[0:i+1], X[0:i+1])
 

 

 
Text(0, 0.5, 'V (m^3)')
 

 

-../_images/bcbf14798c63e658a88847e9ab8d92d2ad5d41762ecc733dc575ba85fcf27e72.png
+../_images/7925349caf05552ea1c4fffd358dc8d04fa332fd610c9b57bd10b7df048e8f32.png

An alternative approach is to use a cumulative trapezoid function. This is defined in scipy.integrate. The main benefit of this approach is that it is faster, as it does not recompute the areas, and the code is shorter, so there are less places to make mistakes!

@@ -1067,7 +1067,7 @@

Numerical quadrature - or integration of functions
0.6666666666666674
-../_images/f706b106a82453b648c11c4cbb788de3b77525a7ab5b3ea707eab348682e497e.png +../_images/4941853b0da2cb3d6a7534e497a1a27a7e15b2dbe146a74f09ba097b53ef9a0c.png

This example is special in several ways:

@@ -1162,7 +1162,7 @@

Numerical quadrature - or integration of functions
-
/tmp/ipykernel_2306/36719416.py:4: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.
+
/tmp/ipykernel_2315/36719416.py:4: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.
   If increasing the limit yields no improvement it is advised to analyze 
   the integrand in order to determine the difficulties.  If the position of a 
   local difficulty can be determined (singularity, discontinuity) one will 
diff --git a/book/03-fode-1.html b/book/03-fode-1.html
index d5f8b41..b84e4ea 100644
--- a/book/03-fode-1.html
+++ b/book/03-fode-1.html
@@ -631,7 +631,7 @@ 
Homogeneous, first-order linear differential equations
 
(array([0., 0., 0., 0.]),
  array([1., 1., 1., 1.]),
- array([4.64672866e-310, 0.00000000e+000, 4.64731861e-310, 4.64731864e-310]))
+ array([4.65721291e-310, 0.00000000e+000, 4.65778914e-310, 4.65778914e-310]))
 

 

 

@@ -744,7 +744,7 @@ 
Homogeneous, first-order linear differential equations
 

 

-../_images/97f47a74e6f5ec6b2c5aa9b558a4ab06bf02bf775e8f4676b0a33ec33fe44090.png
+../_images/b4c4407612597aaf1dc5b7dcd5cdad12ecc8ccc26336b9360c5c547f15f0c75f.png

We should ask, how can we tell this is correct? We can confirm the initial values, which we know are correct.

@@ -775,10 +775,10 @@

Homogeneous, first-order linear differential equations

-
<matplotlib.legend.Legend at 0x7f4078c6c510>
+
<matplotlib.legend.Legend at 0x7f5a95471410>
 

 

-../_images/3b1774314c89078b043bb0c33283a90cd92c482c382ef8c116894bdfecf4db45.png
+../_images/0d992e05e05f92dc53b78d747da1fa725f1cb6fc66247847be3c6e338ba12bc2.png

Here you see good agreement over most of the range. The end-points are always less accurate because the derivatives there are approximated by a less accurate formula. We interpret the sum of this evidence to mean our solution to the ODE is good over this range of x values.

@@ -930,7 +930,7 @@

Euler’s method -../_images/44169436c96c5826b505aba3051d0f793ab0d25d9ec07dae12c8f310f229e586.png +../_images/baf213a87a74d6892adf688865ad1b7d1ae8e9f851fe900bfb0808f194408d31.png

This solution does not look that good until you increase the number of points (i.e. decrease the value of \(h\), significantly). It is known the error decreases only linearly with \(h\).

@@ -978,7 +978,7 @@

Fourth-order Runge-Kutta method -../_images/2473e8c3c25d2f234cd0838a363daeb1f6251898d10a36c55adc0c3ed2cb09f7.png +../_images/f4f1fe9e61c82faa562a6747d282267dc865a7d33a249741dbd0a689b180f1a6.png

Note you can get a much more accurate solution with a larger \(h\) with this method.

@@ -1075,7 +1075,7 @@

scipy.integrate.solve_ivp -../_images/7e2b310a6d6393b989ce6e5a3f4fbf46ffee14ab87557f1a169cf9533fde97f1.png +../_images/4faca789649c75ed68ea65df2851f13de025042d6acaae868c2ba1f38a115875.png

That doesn’t looks so great since there are only four data points. By default, the algorithm only uses as many points as it needs to achieve a specified tolerance. We can specify that we want the solution evaluated at other points using the optional t_eval keyword arg.

@@ -1114,10 +1114,10 @@

scipy.integrate.solve_ivp -
<matplotlib.legend.Legend at 0x7f4078c50690>
+
<matplotlib.legend.Legend at 0x7f5a93130290>
 

 

-../_images/b6e9ecd9195ceba44598f4aed10f470e051415b77dff29d19d311e2b68cf0f66.png
+../_images/0f3cb0f20de8a66563583063f2afeb51738e8b0db5b5ace82e7cb68a5289387e.png

So far, solve_ivp solves the issues with item 1 (we did not have to code the algorithm), and items 2 and 3 (it uses an adaptive step and converges to a tolerance for us). It will also help us solve for the inverse problem, i.e. for what value of \(x\) is \(y=4\)?

diff --git a/book/04-fode-2.html b/book/04-fode-2.html index ed55d73..b17c309 100644 --- a/book/04-fode-2.html +++ b/book/04-fode-2.html @@ -734,7 +734,7 @@

Families of solutions to FODEs -../_images/b63004c7c21c1a661480ce19cc92fe4e776351adb064ae041d737e4ab38d4ea0.png +../_images/4c882170c57edbe19f75eea15d2b78930e473c1428b0eaa0c9444e7eedd340ff.png

If you pick a point, the arrows show you which way the solution goes from there. You just follow the arrows to get an approximate solution to this equation. Let’s consider some specific solutions. Suppose we start with the initial condition that \(y(-1) = 0\). You can trace the arrows to estimate where the solution goes.

@@ -769,7 +769,7 @@

Families of solutions to FODEs
Text(0, 0.5, 'y')
-../_images/86a31f80392852fc21556106f3de0b504ecfa11317ef148be4b65792e8dfd1b1.png +../_images/4ca25257f2913f38ae85d4a95038cebaf45d532e1a32be08ac0b3fcd328a857a.png

Here are some more examples.

@@ -791,7 +791,7 @@

Families of solutions to FODEs -../_images/a1cf9b0c1948b3cdd8aa6a352b85a1df8309149354e392fcfbcc8fbe91286ccc.png +../_images/7a2d3ec2aecbb38990f26d5d568e4ae26ebbf5ff69f71e784a0d7ea0620787b6.png

You can see the solution looks different depending on the initial condition, but in each case the solution follows the direction field.

@@ -812,7 +812,7 @@

Families of solutions to FODEs -../_images/364f9992b8211498defc0934a7f176ac26ebca6c233638d1dc2b6b1282512b6f.png +../_images/e8aa314b47b56d6047d707f881a7fa79a2376278f871025b873b258c92b05dc5.png @@ -949,7 +949,7 @@

Systems of first-order differential equations -../_images/1c3e4587403683649be4c418331b7331ff87488f2ae1f41d456f955aeda179b2.png +../_images/f66d2a7bd23d952241f13cb70f14341eec26d3af9f58a7499afa25a82e55c6be.png

Another way is to convert the solution to an array where the data we want to plot is in columns. We can achieve this by transposing the array to convert it from 2 rows with 50 columns to 50 rows with 2 columns.

@@ -1025,7 +1025,7 @@

Systems of first-order differential equations -../_images/1c3e4587403683649be4c418331b7331ff87488f2ae1f41d456f955aeda179b2.png +../_images/f66d2a7bd23d952241f13cb70f14341eec26d3af9f58a7499afa25a82e55c6be.png

This works because you can plot an array where the values to be plotted are all in columns.

@@ -1089,7 +1089,7 @@

Predator-prey model example -../_images/5b2910e270ab7d763061402a1308f6b1a0c5ca6f2cebac6f9f713a6f63ea2cd3.png +../_images/ac097c2a6c4aaf204819d804907750128d617a66d53359627cb7842e30ad6445.png

This is a classic boom/bust cycle of predator/prey.

@@ -1120,7 +1120,7 @@

Qualitative method for systems of ODEs -../_images/1f5542999a1f5d8c434daa8f9e0e0f4aa810a426acd91647ddc11cb4c21c6416.png +../_images/c7418030fbdc457831f46f2b2f697082aba6bf810cf31ff06f1e083398379356.png

In this view, we have a limit cycle which just shows the number of rabbits and foxes goes up and down periodically as you travel around the solution curve. Time is parametric in this plot. It starts at t=0 at the initial state, and increases as you go around the cycle.

diff --git a/book/05-nth-odes.html b/book/05-nth-odes.html index e490651..c92ef8e 100644 --- a/book/05-nth-odes.html +++ b/book/05-nth-odes.html @@ -623,7 +623,7 @@

The Van der Pol oscillator -../_images/f111a77b595e89cdb2e29f33eacd6bafa1f35453167223420cdde9cc2521c419.png +../_images/d7b2ba1a939043517519991ed0de1a12fa3ee661d39555354f670a111932217b.png

You can see that the solution appears oscillatory. Let’s be more quantitative than what it looks like. An alternative way to visualize this solution is called the phase portrait where we plot the two state variables (x, v) against each other. We include the starting point for visualization.

@@ -638,7 +638,7 @@

The Van der Pol oscillator -../_images/8ae4c21f593761bc397a8de7102ec9d088ecacb8a895b20c29334dc840432519.png +../_images/1bfbac7247a457c7f1a00ae8fdfd364d165de42d68e127c9ef0b537813b441ad.png

So, evidently it is not exactly periodic in the beginning, but seems to take some time to settle down into a periodic rhythm. That seems to be the case, because if it didn’t we would expect to see a continued spiral in or out of this limit cycle. Another way we can assess this quantitatively is to look at the peak positions in our solution. We return to an event type of solution. We seek an event where the derivative \(dx/dt=0\), and it is a maximum, which means \(x'\) starts positive, becomes zero, and then is negative. Note this is appropriate for this problem, where there is only one, periodic maximum. For other problems, you might need a different approach.

@@ -703,7 +703,7 @@

The Van der Pol oscillator -../_images/83afda2003392c55fb4c60b8c8829bd537a8016b4cd059feb2601929e1dc6d80.png +../_images/33392e0637fdc9771df76c1fb7e43db7e7687a6ddf08c2d66e24180292372af0.png

That looks good, the red dots appear at the maxima, and they are periodic, so now we can see how xmax varies with time.

@@ -716,7 +716,7 @@

The Van der Pol oscillator -../_images/5473213f300a06baf858975406f8d2b754c1fd61822a030979c83b002b923f82.png +../_images/32b0a7e5ccd0208675f3f666e5cd100d653a9aa3d16e154a3a63f8048010809f.png

You can see that after about 5 cycles, xmax is practically constant. We can also see that the period (the time between maxima) is converging to a constant. We cannot say much about what happens at longer times. You could integrate longer if it is important to know that. This is a limitation of numerical methods though. To prove that it will be constant, you need to do some analytical math that would show the period and xmax go to a constant.

@@ -729,7 +729,7 @@

The Van der Pol oscillator -../_images/97fd073914199533ea8be1896f2f10d13331fb647bb4f98f12528d74b9a9bf30.png +../_images/029b1bcc3bc1e118020ecc7c9951ae4a06c452a98e60b2051d5ad6c0f5bbe090.png

If we seek the steady state, oscillatory behavior of this system, we should discard the solutions in at least the first 4 cycles, since the maxima and periods are still changing.

@@ -760,7 +760,7 @@

The Van der Pol oscillator -../_images/22a17bd10012654af9a38f44624f3f968285839fabaa68a9195e4c6838012d28.png +../_images/2d2de64a3539ac993dd5d9123a719e8b69cc398c4eaff125d285fcb41736daff.png

Here you see about 6 more cycles. The period of these events is practically constant.

@@ -789,7 +789,7 @@

The Van der Pol oscillator -../_images/f84c53b555a07e9d452de4273efaa3253185e8bc9c8fe13843550aa1851662f7.png +../_images/1c432c090e842a53aed8ae1a8f2a9c20353b4783e0fa0604c14957350fe81a81.png

This limit cycle shows the oscillatory behavior. You can see here that each cycle repeats on top of itself.

@@ -907,7 +907,7 @@

Solving a parameterized ODE many times
For mu=0.01, steady period after 0 iterations
-../_images/a5efe01b25760247dcac46de8444ed86f888152b33da77f2fea30e1bddb890fb.png +../_images/8e4b2231cf1a138a6bf3e8d27860212b9df390ca9070a4ba85a7e6922f059a56.png diff --git a/book/07-nla-1.html b/book/07-nla-1.html index 8b6878b..057a4c6 100644 --- a/book/07-nla-1.html +++ b/book/07-nla-1.html @@ -604,7 +604,7 @@

Introduction to nonlinear algebra -../_images/30d9337ce50a61d8227da4e3127520aa9e7fd1f390181991b9af33066510a1e6.png +../_images/fbf1086e30e4b545b78502de84313b8fc260376fc3e8cf3bbefb769666105d60.png

In contrast, \(sin(x) = 0.5\) will have an infinite number of solutions, everywhere the function intersects the x-axis.

@@ -623,7 +623,7 @@

Introduction to nonlinear algebra -../_images/60ee5d949cd85eff18f192cfcb947f1d3a47a69852928d703de6f14d82146e0d.png +../_images/066977168613b28fa6fac9b75bb25dd3d2c9a1db5162c26b8e45c4ceb1d470f6.png

Finally, \(sin(x) = x - 1\) has only one solution.

@@ -640,7 +640,7 @@

Introduction to nonlinear algebra -../_images/6a83fbff77d4cbfd129ab40fc8e80a93e12da822ff5e234ff739b822296141c3.png +../_images/2405a236dcb32f376d0aabea00b08253c705780d75c6ade473cf06e92b8bcb2a.png

The equation \(e^{-0.5 x} \sin(x) = 0.5\), evidently has two solutions, but other versions of this equation might have 0, 1, multiple or infinite solutions.

@@ -660,7 +660,7 @@

Introduction to nonlinear algebra -../_images/8102e3c62117d467f06be43ac820095eff2629f59d961d86453636b1cd730473.png +../_images/107efcebc94ee8b84ea73a5d509e0a4aa1c7811487e938724c88e0864dbd4246.png

exercise modify the equation to see 0, 1, many or infinite solutions.

@@ -783,7 +783,7 @@

Problem problems -../_images/f69fab6596408746dd18c4800abc16ed82e584aa9b119fa3293d3d7c10aa8f3c.png +../_images/81ff21e0d7623df262b9c4623435236b76e36b0fc73c9b984db17149300f0acd.png

It seems obvious there is a root near -1.7. But if you use a guess around x=0, the algorithm simply oscillates back and forth and never converges. Let’s see:

@@ -854,7 +854,7 @@

Derivatives of functions -
/tmp/ipykernel_2413/1845505028.py:6: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2430/1845505028.py:6: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   derivative(f, x0, dx=1e-6), 3 * x0**2  # the numerical and analytical derivative
 

 

@@ -899,9 +899,9 @@ 
Derivatives of functions
-
/tmp/ipykernel_2413/1563071673.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2430/1563071673.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   d0 = derivative(func, x0, dx=dx)
-/tmp/ipykernel_2413/1563071673.py:10: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+/tmp/ipykernel_2430/1563071673.py:10: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   dnew = derivative(func, x0, dx=dx)
 

 

@@ -937,9 +937,9 @@ 
Derivatives of functions
-
/tmp/ipykernel_2413/1563071673.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2430/1563071673.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   d0 = derivative(func, x0, dx=dx)
-/tmp/ipykernel_2413/1563071673.py:10: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+/tmp/ipykernel_2430/1563071673.py:10: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   dnew = derivative(func, x0, dx=dx)
 

 

@@ -982,7 +982,7 @@ 
fsolve#

 

-
/tmp/ipykernel_2413/1619064557.py:9: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+
/tmp/ipykernel_2430/1619064557.py:9: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   print(f'{float(ans):1.2f}')
 

 

@@ -1146,7 +1146,7 @@ 
A worked example
-../_images/e92bb1557ffe92211c1640ac76cc40e009bac044d4ac0c207b95a4d778d4d1a9.png
+../_images/fdc01fbc59c1f2cb1d247a6f579dc0be1667f33874af3a1850315b3d2b230e7f.png
 

 

 
You can see there is one answer in this range, near a flow rate of 1.0 mol/min. We use that as an initial guess for fsolve:

@@ -1212,11 +1212,11 @@ 
Parameterized objective functions
-
/tmp/ipykernel_2413/2335501430.py:17: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
+
/tmp/ipykernel_2430/2335501430.py:17: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
   fa_exit[i] = ans
 

 

-../_images/cd2f0004d3d8d62f7720750098ee15a07d42e86d421e7e5cbd8e4d30235bba58.png
+../_images/aa84161603fd8ceeb44ebc38bb6ae065bdd634223437cd1e21e504037785a6ce.png
 

 

 
You can see here that any rate constant above about 0.5 1/min leads to near complete conversion, so heating above the temperature required for this would be wasteful.

diff --git a/book/08-nla-2.html b/book/08-nla-2.html
index 2be6250..1c6eb42 100644
--- a/book/08-nla-2.html
+++ b/book/08-nla-2.html
@@ -620,7 +620,7 @@ 
Special nonlinear systems - polynomials
-../_images/740391898af766dce518adaca306e6844e20d097e2735c681d0a3977ddfd8597.png
+../_images/30f1a9ae200d2cdc4c033a74b4a2315451a3baf6f88e5149995d0b7cee272f88.png
 

 

 
We can also use this to plot a polynomial.

@@ -639,7 +639,7 @@ 
Special nonlinear systems - polynomials
-../_images/24dd69271dda5347b85b2b103f6f0782c7796148fe9583efda5bbff01bf06b5f.png
+../_images/9936a95a3228c3b18c476401e29db943acde4b888f24c04629179410406d249c.png

Why is this so convenient?

@@ -721,7 +721,7 @@

Cubic equations of state -
/tmp/ipykernel_2438/406724879.py:1: ComplexWarning: Casting complex values to real discards the imaginary part
+
/tmp/ipykernel_2462/406724879.py:1: ComplexWarning: Casting complex values to real discards the imaginary part
   float(R[0])
 

 

@@ -799,7 +799,7 @@ 
Other useful things to remember about polynomials
-
/tmp/ipykernel_2438/174692805.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2462/174692805.py:3: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   np.trapz(Y, X)
 

 

@@ -836,7 +836,7 @@ 
Other useful things to remember about polynomials
-
133 μs ± 606 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
+
130 μs ± 306 ns per loop (mean ± std. dev. of 7 runs, 10,000 loops each)
@@ -853,7 +853,7 @@

Other useful things to remember about polynomials

-
6.37 μs ± 21.5 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
+
6.34 μs ± 105 ns per loop (mean ± std. dev. of 7 runs, 100,000 loops each)
@@ -885,7 +885,7 @@

Other useful things to remember about polynomials

-
<function integrand at 0x7fb934925a80>
+
<function integrand at 0x7fc7bc42b880>
@@ -980,7 +980,7 @@

Systems of nonlinear equations -../_images/b21e5afe6d9c40bef6fbd2f854ce9a8239c9131aa61506b578856c2008f72857.png +../_images/cfd3a55c15f58b6dc7388b5cdd9a5de2a3a277010f8a07aecda067e0fcf3dfd0.png

You can see that on this domain, there is one place where the two curves intersect near the point (2, 5), which is a solution point. At this point there is one (x, y) pair that is a solution to both equations.

@@ -1033,7 +1033,7 @@

Systems of nonlinear equations -../_images/571c4480d503f03e05e5d947d1f76e76c3ed3364f4ed646f924e58aa370e06bc.png +../_images/baa658d89c12b9a6874804de33737774032446822a064bec6ff49d28bef5f196.png

@@ -1047,7 +1047,7 @@

Systems of nonlinear equations -../_images/8d5cbac746adba42daf1cbff45da7cf2716084781c336455806affa3ea5a74f9.png +../_images/a9d29d143afc1478b34fc5dce21d0d7d8af6bae3cb8c29da2a0280663a38cf6d.png

There is an intersection near \(x_1=0.4\), and $x_2 = 0.6. We can use that as an initial guess.

@@ -1077,7 +1077,7 @@

Systems of nonlinear equations -../_images/c3894a8cc018e12821fff5bda2d642bcb4f1b924ca2633fa1fc5f6195bf700ad.png +../_images/f1696445d6889128b38f3b745598828fc15e4da24dbb2f615d9f2dd6458a781d.png

@@ -1095,19 +1095,12 @@

Systems of nonlinear equations -
Failed at guess [-0.38179900425446167, -1.0685278685719757].
+
Failed at guess [9.553783369922858, 6.704263032371085].
 The iteration is not making good progress, as measured by the 
   improvement from the last ten iterations.
-Failed at guess [9.0839411809098, -3.5346006932896].
-The iteration is not making good progress, as measured by the 
-  improvement from the last ten iterations.
-

-

-
/tmp/ipykernel_2438/476845304.py:3: RuntimeWarning: overflow encountered in exp
-  z1 = np.exp(-np.exp(-(x1 + x2))) - x2 * (1 + x1**2)
 

 

-
array([3.62834901e-11, 6.44142390e-11])
+
array([6.25721551e-12, 1.18666138e-11])
 

 

 

@@ -1122,7 +1115,7 @@ 
Systems of nonlinear equations
-../_images/3405259dd602934321ae9f74af0281ce280e1793e32729fe5d21680a8a47fea8.png
+../_images/726aa73f113a83473a59d51fe4c145d62b872eb49215e5ff1cf3ca8ea5a6f487.png

This shows the solution, and that the objective is practically equal to zero at that point.

diff --git a/book/09-bvp.html b/book/09-bvp.html index 80574d2..b946054 100644 --- a/book/09-bvp.html +++ b/book/09-bvp.html @@ -706,7 +706,7 @@

Solving nonlinear BVPs by finite differences -../_images/19b427ab270dd74712aab0075982c03c42b93d7ac3f7e0efb28793348047f701.png +../_images/cb6daf9eca181f6de520eafc1dad5323124f52a7d5eb40782e301060d6cf8b8b.png

We should check our residuals function. We mostly want to see that it runs, and produces the right shaped output.

@@ -738,7 +738,7 @@

Solving nonlinear BVPs by finite differences
The solution converged.
-../_images/7b9ff5f7e68229fd5c53414fa7a98f28b1641912bf5bca142a99c99cb439b4d7.png +../_images/dae5ce25a27aa9e40f3806b2229e1755b9da44b9fe83ee1e74cba2cd5b9ec68d.png
@@ -779,7 +779,7 @@

Solving nonlinear BVPs by finite differences -../_images/29fa31f0add328491fc65f156f2e24dd0efc5fbcc54c81aae45d0894ba7ec8d3.png +../_images/8700d108e681f489df1577e24c1bd862ea39ac55b6ee79ab139e476fd4ebb0c6.png

This result doesn’t look great at the origin, but remember:

@@ -882,7 +882,7 @@

A worked bvp problem
((50,), (50,))
-../_images/8fe14c22db72781a4c49634c655235ea02d020f43f33868a7e11537ae97dcf9a.png +../_images/5937a4f15ecc476ae85579d1ba6305d49a86c885dcdb7538933d73fe0d4ace5e.png

We also need a guess for U2, and in this case we know that \(u2 = u1'\), so we just use that.

@@ -903,7 +903,7 @@

A worked bvp problem
(2, 50)
-../_images/1f218fe8bf6df35223ba30686fccfc17b1ff1e9eabf2f598f8a6413b54bd1075.png +../_images/1a98bbe977711241a49194c6852a5b7410a6db551d1e330f0d062f976bbc955c.png

You should always visualize the guess to make sure it does what you want. It is hard to make these!

@@ -922,7 +922,7 @@

A worked bvp problem
-../_images/fd1cfa9c8cc1ca9e128be7d0c06066b44410c2b711e1fb35b734e7ddf2788c73.png +../_images/f76a8c91804911e9579d8c1a2b5ff873b9c204ccce46e5752f6993047cd3f62b.png

Now, we are ready to solve the BVP.

@@ -942,7 +942,7 @@

A worked bvp problem
The algorithm converged to the desired accuracy.
-../_images/c9138b755f288ca87f2393af6b9fc51b153260282ba5cbf36c216697e6e4b885.png +../_images/20b752d591048727da1fa7d833b602535a309d7ee2a3339d304c5611b8988010.png

exercise Try using different guesses, e.g. lines, or triangle shapes, etc. What else looks like this shape? Half a cycle of a sin wave? A semi-circle?

@@ -995,7 +995,7 @@

Concentration profile in a particle -../_images/94974a1e90dfed707653221a5252e77b14d01273eef933af51dcb41699478165.png +../_images/38ab6e4fc7d18a4ad04fd8e9ef8f582c184bb90759fb16a959c9986bda1be8ea.png

Now we solve the system.

@@ -1028,7 +1028,7 @@

Concentration profile in a particle -../_images/09599c841a8337fd5122990ae72614ea743dd9945aa28c48e4852a7541852191.png +../_images/111f6c2981b5ffbd33250e61e82fb720d0b348723a1f76a0136775fddea229c5.png
@@ -1042,7 +1042,7 @@

Concentration profile in a particle
50

-../_images/3763b4458b314e06dfb7f72273c3e69037c310c9f0aa0cd2ed015f8f4139e809.png +../_images/ecf2bf2691ae87ca76cc07d3865ad9c991553ea4e79403e35c52db47e1d1ce99.png
@@ -1076,7 +1076,7 @@

Concentration profile in a particle
0.008249311811255714

-
/tmp/ipykernel_2469/1627362779.py:2: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2497/1627362779.py:2: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   print(np.trapz(c**2, sol.x))
 

 

diff --git a/book/10-min-max.html b/book/10-min-max.html
index d3e74d6..b9e767e 100644
--- a/book/10-min-max.html
+++ b/book/10-min-max.html
@@ -597,7 +597,7 @@ 
Function extrema
-../_images/5a927867c3b4582ca1c3dadd1f9719999a9944fc7f4c0fbad3694bcf47c1ec98.png
+../_images/f68a5156693e14f9277e8f1267cdef083ad3a4c67767f199aca4abe22f50964d.png

You can see there is a minimum near 0.6. We can find the minimum in a crude kind of way by finding the index of the minimum value in the y-array, and then getting the corresponding value of the x-array. You control the accuracy of this answer by the number of points you discretize the function over.

@@ -664,7 +664,7 @@

Find the derivative, and solve for where it is zero
-
/tmp/ipykernel_2496/2917585576.py:4: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2522/2917585576.py:4: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   return derivative(f, x, dx=1e-6)
 

 

@@ -707,9 +707,9 @@ 
Newton-Raphson method of minima finding
-
/tmp/ipykernel_2496/4103338182.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2522/4103338182.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   yp = derivative(f, x0, dx=1e-6, n=1)
-/tmp/ipykernel_2496/4103338182.py:8: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+/tmp/ipykernel_2522/4103338182.py:8: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   ypp = derivative(f, x0, dx=1e-6, n=2)
 

 

@@ -776,7 +776,7 @@ 
scipy.optimize.minimize
 

 

-../_images/4e3a24f53230e78e938c5ab0c7572675c7784cf0901013467e43667d433caa41.png
+../_images/e5cfc94592a6a5793750e200b13728dd24f26de11b636b5e0c9cd9a046fe7694.png
 

 

 
Note this answer is only the same in the first 4 decimal places. Remember that these iterative approaches stop when a tolerance is met. Check the defaults on fmin!

@@ -797,7 +797,7 @@ 
Multiple minima
-../_images/69add91c1d95a7aca8da7c7199a16199c1c45236fe2260fd4e07ad236c969975.png
+../_images/78b3c1e9567d501e2f042a6fe423603d051c490e9b4d30c79ca95cc7b26d595e.png

This guess finds the one near 2.2:

@@ -858,7 +858,7 @@

Finding maxima -../_images/1944484f5a832cf925b3b378624e4dfaf461406c7f6cc6b8a4d0afcf0fa44d05.png +../_images/993ee85392d96f8332d25b370271990469b82a0111cd8a0c62115911f5d1d8c8.png

The standard way to use fmin is to define an optional argument for the sign that defaults to one. Then, when we call fmin, we will pass -1 as the sign to the function, so we find the minimum of -h(x). Then, we evaluate h(x) at that x-value to get the actual value of the maximum. It is not necessary do this, you can also manually pass around the sign and try to keep it straight.

@@ -882,7 +882,7 @@

Finding maxima
[1.56120872]
-../_images/eed310bec70a07afa3a8201dab33583a44345de12e5cfda22517d2f2f6049c1c.png +../_images/4dfae24e648ab1ff919e9ce216b1a8045591666e680eb78d89f41b05a8b4c4a8.png

Once again, here you have to decide which maximum is relevant

@@ -955,7 +955,7 @@

Application to maximizing profit in a PFR -../_images/bd7b31a74bd168e63b7a7077c6c0b577ea0e68efa78f98be76151a8b01826796.png +../_images/35aecf9687338b391e751883ca2d78d8b2afe018c5ba017780b0b4add7ac514c.png

You can see from this plot there is a maximum near V=1.5. We can use that as a guess for fmin.

diff --git a/book/11-regression.html b/book/11-regression.html index 380aee2..df03d5b 100644 --- a/book/11-regression.html +++ b/book/11-regression.html @@ -593,7 +593,7 @@

Regression of data is a form of function minimization
-../_images/b1b10890e1cde814b50f99018fdc9419242834bb2c61f44961df167b2b6ec874.png +../_images/b5f71e60a1ff65c161c08c0675cf07c52c8d94843a3840cbc983c4d6da386433.png

In Materials Science we often want to fit an equation of state to this data. We will use this equation:

@@ -746,7 +746,7 @@

Regression of data is a form of function minimization -../_images/388fd650fb0d5d273c44562261f06b735bf19429fb10d7188ccdd970104a948f.png +../_images/bfcfe201836f5d4df7022a1e96e59017620d4e880f439c428eb33ec328a52e69.png

That looks pretty good. We should ask ourselves, how do we know we got a minimum? We should see that the objective function is really at a minimum for each of the parameters. Here, we show that it is a minimum for the first parameter.

@@ -766,7 +766,7 @@

Regression of data is a form of function minimization
-../_images/fa0bceee4a68863bf8524bc64cd409d572cc7d961f8fc0f63bdfd50dfbe90d7a.png +../_images/9d7b6ace81d9855f16bad8b5fdc65362401f029dd9cddc8debac3390bef3e265.png

You can see visually that the error goes up on each side of the parameter estimate.

@@ -818,7 +818,7 @@

Regression of data is a form of function minimization
-
/tmp/ipykernel_2522/3278399195.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
+
/tmp/ipykernel_2548/3278399195.py:7: DeprecationWarning: scipy.misc.derivative is deprecated in SciPy v1.10.0; and will be completely removed in SciPy v1.12.0. You may consider using findiff: https://github.com/maroba/findiff or numdifftools: https://github.com/pbrod/numdifftools
   dEdV = derivative(proxy, V, args=(pars,), dx=1e-6)
 

 

@@ -895,7 +895,7 @@ 
An example with curve_fit
-../_images/4aa8675679ec8734d160ef99900f12a82f6efba5724f31f8a4e54b9a98b5cbd2.png
+../_images/b751c877d5d573cd904cfbd848b7d7a2cd83797d9278a713dcd532015c75dbb2.png

What should we use for an initial guess? At \(x=0\), \(y = 0\), which isn’t that helpful. At large \(x\), we have \(y=a\). From the data, we can guess that \(a \approx 1.2\). For small x, we have \(y = a/b x\). So, if we estimate the slope, we can estimate b. We arrive at these guesses by thoughtful inspection of the data, and the model that we use to fit it.

@@ -964,7 +964,7 @@

An example with curve_fit -../_images/7a00690216e766e05ae329b444a2e0e04d50659e88c64644336542c9f1cf655c.png +../_images/a57dee269eaa0ffafcd7c429127387eb43117e954db303c455f3c326bef3fe9c.png

exercise Try different initial guesses and find one that does not look this good.

@@ -1097,7 +1097,7 @@

What about uncertainty on the predictions?
np.float64(1.3205427044923441)
-../_images/5d69ed60d0d6b73f00b0e4c891cc50f74c9aa4dbb5511e9b476caeafdc41375e.png +../_images/bf798e624d76d3c8e1cde5cbdd6676700cebfaa78edcce91cbdef71bc4f23742.png

We estimate the model plateaus at about y=1.32, but what is an appropriate estimate of the error in this? There are uncertainties in the model parameters, so there must be uncertainty in the predictions. To estimate this, we first look at how to generate a distribution of random numbers with a normal distribution around some mean with some standard error.

@@ -1112,7 +1112,7 @@

What about uncertainty on the predictions? -../_images/57a592bc4966bce3f64dcb4976c8b9237d2f79bd8b8c26a0d952f3f00961612e.png +../_images/824a708ef3594117a584e21306433169b31dfb0c126fc72b59caeda6b5d040eb.png

So the idea is we can generate a distribution of the parameters

@@ -1128,12 +1128,12 @@

What about uncertainty on the predictions? -
(np.float64(1.3204388883201963),
- np.float64(0.009640609264270194),
+
(np.float64(1.3203509874995711),
+ np.float64(0.00962727705451916),
  np.float64(0.001027904909551584))
 

 

-../_images/c70ab1126da7211c94f525ba62f0277b377f124efad5deb369b27e7e97eb913e.png
+../_images/2a627a3dc4f1ed5e7987361c98a7e69502d3d4d7d0da4b48e492c3a83b371d1d.png

Well, in 20/20 hindsight, we might have guessed the uncertainty in the asymptote would be just like the uncertainty in the \(a\) parameter. In this case, it is appropriate to use three significant figures given the uncertainty on the answer. A useful guideline is that the 95% confidence interval is about ± 2 σ. At ± 1 σ you only have about a 60% confidence interval.

diff --git a/book/12-nonlinear-regression-2.html b/book/12-nonlinear-regression-2.html index 0da2e8f..86a987a 100644 --- a/book/12-nonlinear-regression-2.html +++ b/book/12-nonlinear-regression-2.html @@ -710,7 +710,7 @@

Effects of outliers on regression
[0.2404 1.144 ]
-../_images/92bedcb7f7efba501510588d5876c7cf2db61faaaa52c12dc645928f5de3b39a.png +../_images/ea2642bee61dda6e8bde49423fa9246377b3ebfcd6edc10a36f3490831fa93ea.png

You can see that the fitted line is “dragged” towards the outlier. We say that least squares minimization is not robust to outliers.

@@ -771,7 +771,7 @@

Minimizing the summed absolute errors
[0.26845682 1.14      ]
-../_images/82f090990f950e3316dd3b3124f1f79a54f990e46c92d451a306f0e410360f76.png +../_images/b74eb817aa063dc63c2101786526877b3831460e8846d12ccd3e5efda0d764a7.png

There is a historical reason this is not done a lot, and that is the absolute value function has a discontinuity in its first derivative at the origin which can be problematic in some optimization algorithms. It is obviously not a problem here, and you can see that the outlier has less of an effect on the fitted line in this case.

@@ -796,7 +796,7 @@

Minimizing the summed absolute errors
[0.25741034 1.15352086]
-../_images/69c8a9e9b5fcefe1846a24a08870939fd87a2d4567d1d977d79bcc2c747db5ec.png +../_images/cce6250ebc7f4466fab3031bf80241bcc1ed7464185d6c6f03635aef3b6cf609.png

The downside of these approaches is that they complicate the analysis of uncertainty. The uncertainty analysis we have considered so far is only formally correct when we minimize the summed squared errors. It is only approximately correct when something else is minimized.

@@ -819,7 +819,7 @@

Robust regression approaches -../_images/336656425b11fe2498d3d211480bc87d6e1c3304f48f736cf9bcc48da0aedc60.png +../_images/6215fa451250e6130f3ac8916075fca377232d20667e8e778d25e6fff00363df.png
@@ -842,7 +842,7 @@

Least Median regression
[0.26804924 1.18981534]
-../_images/431922bc7cce979c7ce5472668718b860fe871446144b2b0d0e8e56c413f7d31.png +../_images/a1e50f72bc5e3e5c0bb353a9d1e93896ed929fda57d1248d914ac7ea47485f81.png

@@ -864,7 +864,7 @@

Weighted nonlinear regression -../_images/e3a06fa40efcf6e7a3f62a3e2e24dd25279f6da1db1ed735c287a7e94fa82a53.png +../_images/fe4f05fe2b33accf2e411ba6247288aacce0f06c210767be95fdae6cef1a2898.png

The aim of this work is to fit a nonlinear model \(y= a (1 - e^{-b x})\) to this data. We first consider a standard minimization of the sum squared errors. Inspection of the model suggests at large x, \(a\) is a plateau value, which we can read from the graph. For the value of \(b\), we might estimate a half-life at about one day and solve \(110 = 240(1 - e^-b)\)

@@ -908,7 +908,7 @@

Weighted nonlinear regression -../_images/bd712f74b2cca54b5f8347da9641be6b9318b46cbb31adcd79bc0e8c3d101c82.png +../_images/c82737ee384e7c49b4f7d9545851b464a090caa76ee8ab9306b223f11ab1a0a1.png

The fit generally goes through the data, but it is not clear if there is a small outlier near 2 that is skewing the fit, and perhaps leading to an inaccurate asymptote at long times.

@@ -942,7 +942,7 @@

Weighted nonlinear regression
[230.77020888   0.35563066]
-../_images/5aa17a10decbe9ebc61b5950330b8370a00d61241698f651ea0f950f66c8f4f8.png +../_images/9eb7c49f28c4ca72a093d4df6a935ace28122f279f9e5fb022a4d9d8fca28804.png

The result here is that the model fits the points we measured a lot better than the points we measured once.

diff --git a/book/13-constrained-optimization.html b/book/13-constrained-optimization.html index 1a88a69..fb915a9 100644 --- a/book/13-constrained-optimization.html +++ b/book/13-constrained-optimization.html @@ -608,7 +608,7 @@

Constrained minimization -../_images/875ba8db0b3560f8ca3dcaa8440412d855810948b0e87c7170610199edf965eb.png +../_images/9470eb37f895154b86414647d12f817bc6c35427b8ba7a0fa3c37d7918b53464.png
@@ -662,7 +662,7 @@

Constrained minimization -../_images/2a4ca56065bfd06d8d1aa09d82112b6db5fc9a3724d2c61c2daab9ed63c118a6.png +../_images/aadd760b2960a2f713ed4721fd48f16008e07e38983876c928d06212c651beae.png

@@ -909,7 +909,7 @@

Equality constraints

-../_images/79eeed0e914677aec38793afd3ac55751b308ce81aa2318d3769fdfa0df3fc85.png +../_images/913e5b378f7e614043547dfcccef60ce11ae7a3b6d6f64719939a542a4fa7ca7.png

You can have multiple equality constraints, you just make a list of dictionaries. Suppose we seek to minimize \(x1 + x2 + x3^2\) subject to the equality constraints \(x1=1\), and \(x1^2 + x2^2 = 1\). Some analysis suggests that this really means x1=1, x2=0, and then x3 must also be zero to minimize the function, which has a minimum value of 1.

@@ -969,7 +969,7 @@

Inequality constraints
-../_images/1c057799368cd45f7f887062dad2ff6f8e28a16e61c5960f8b0e14d3010324f1.png +../_images/53ac317667d7c2fc20cbd3e3c4aacc09ab5cacd9ad0ad1bd1f621b38162a18e3.png

You can see by inspection there is a minimum around x=-1, and at x=2.5. Note the one at x=2.5 is not a minimum in the sense that the derivative=0 there, it is just the smallest value that also satisfies the constraint. To solve this problem, we set up the following code:

diff --git a/book/15-intro-linear-algebra.html b/book/15-intro-linear-algebra.html index 4f622dc..691e026 100644 --- a/book/15-intro-linear-algebra.html +++ b/book/15-intro-linear-algebra.html @@ -1257,7 +1257,7 @@

The determinant -
np.float64(-9.646182399139509)
+
np.float64(-0.21806302411547418)
@@ -1276,9 +1276,9 @@

The inverse -
array([[  7.19524034, -18.73469219,   3.8115125 ],
-       [-10.13704231,  25.94168394,  -2.92184225],
-       [ -0.62448163,   7.82010496,  -2.83597352]])
+
array([[-0.98453046, -0.01633357,  1.57391102],
+       [ 0.49000804,  2.23325752, -2.34893843],
+       [ 1.23665877, -1.39040644,  0.70156965]])
@@ -1292,9 +1292,9 @@

The inverse -
[[ 1.  0.  0.]
- [-0.  1. -0.]
- [ 0.  0.  1.]]
+
[[ 1. -0.  0.]
+ [-0.  1.  0.]
+ [-0. -0.  1.]]
@@ -1569,7 +1569,7 @@

Solving linear algebraic equations -../_images/6e5edd7881b60220a6769eedcf205c10621a625cc4b1c1e075f2899c8fd74c3a.png +../_images/8f479c25eb372fea1e52cb55052a91f598cf96f897aa18d26caf2ca7d87bddaf.png

As usual, there is a function we can use to solve this.

@@ -1604,7 +1604,7 @@

Solving linear algebraic equations -../_images/2b6e6661d070ec6f6aaa4b7825aaf71d2b9ee517a16c2dfb0425fcbea66c864c.png +../_images/6c66440a130e6d454353af7eb8bd3305469d07e626e17e421dc8229fc118e04c.png

You can see by inspection that solve must not be using an inverse to solve these equations; if it did, it would take much longer to solve them. It is remarkable that we can solve ~5000 simultaneous equations here in about 1 second!

diff --git a/book/16-linear-algebra.html b/book/16-linear-algebra.html index 439264d..d90cf29 100644 --- a/book/16-linear-algebra.html +++ b/book/16-linear-algebra.html @@ -1029,7 +1029,7 @@

Application in linear boundary value problems -../_images/6194abc785c67cc46f00b47baa4e0fc39849586702a5b5412fe232169097c56f.png +../_images/c2be06cca0f7bfc352dad9c6c6c9c52bd7a66ebd7b70df92aee2d4ef31597b52.png

Note that we have approximated the solution by discretizing and estimating the derivatives that the points. You have to check for convergence by increasing the number of points \(N\).

@@ -1108,7 +1108,7 @@

Things to look out for
-../_images/7c8ce8739b38a423088537f48abfdef6a4a5f01c55439bfed4eeca5a8a7037b4.png +../_images/f77ee51b7ce73b2c66aa675dab0a27339ca7cbd2989d943061c78bf2c600c172.png

This system of equations is sensitive to roundoff errors, both in the coefficients of \(\mathbf{A}\) and in the numerics of solving the equations.

diff --git a/book/17-linear-algebra-2.html b/book/17-linear-algebra-2.html index 773c0e7..0779af5 100644 --- a/book/17-linear-algebra-2.html +++ b/book/17-linear-algebra-2.html @@ -594,7 +594,7 @@

Interpolating between data points -../_images/7e317b3c007f55f4a255d368ab829c7bd4ee69652a9aa5ba11abd9f66a3574f7.png +../_images/94979f45a163837a8d61d8abec7c8946c8445bc362ce58a850d0c2926ee1ebd9.png

We would like an equation like \(y(x) = a_2 x^2 + a_1 x + a_0\). If we write these out for each data point we get:

@@ -656,7 +656,7 @@

Interpolating between data points -../_images/a7e500235f3c1e301f3fdbb82a3444fbb5b8777cb368018c14c0829ea7514119.png +../_images/75ecea12fd7cc49008910b1f7001227e678ce9eb6654dd7ca4eab6cb34241fda.png

What we have done here is fit an Nth order polynomial to \(N\) data points. There is a possibility that we have overfit this data, and extrapolation is not reliable. However, interpolation by this method may be useful. We will return to this for larger data sets where \(N\) is much larger than the order of the polynomial when we talk about linear regression next week.

@@ -707,7 +707,7 @@

Interpolation libraries -../_images/2cc78b67ba659d267dab63b42844535e1f69ec5f52d4123f405c02aa96aa9466.png +../_images/3320417e17bb9d94388c5d2bd083d90c29dc3b86eb5cb7dded81dbb0790681f2.png
@@ -721,7 +721,7 @@

Interpolation libraries

-
/tmp/ipykernel_2673/3098855599.py:5: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2692/3098855599.py:5: RuntimeWarning: The iteration is not making good progress, as measured by the 
   improvement from the last ten iterations.
   fsolve(obj, 7)
 

@@ -741,7 +741,7 @@ 
Interpolation libraries
 

 

-../_images/02a28556cba062387a078fa8d04b482dd05eb2d32699473e94f0b01c9674951a.png
+../_images/042636fc8afc51e36797385046aefd10e0d0e60bc2871f54eb6aad210c9dd0de.png

With more data points you can also use cubic interpolation, which fits a cubic polynomial between the points, and ensures smoothness and continuity of the derivatives at the endpoints.

@@ -760,7 +760,7 @@

Interpolation libraries

-../_images/f7c732da14289bc90467d52a702d13b95ff066c91e147ad15fe13401755d9241.png +../_images/42ad92f8432b9e8a1b154f76e088444a4331d83c44a5ffdbec80ee823bf89536.png
@@ -775,7 +775,7 @@

Interpolation libraries

-../_images/808b7664b8358dc80e1f0f0e01e3c92f9e65c42026d04312fe9c901462b9b964.png +../_images/4f1d7d436e7583794320696ee725114d155b62aeba918839f8de5b0ef258e44f.png
@@ -789,7 +789,7 @@

Interpolation libraries

-../_images/64afd8fbcbb2ce8686c0b80db379306bd3b8a72f83b07409a83b46b331a3b846.png +../_images/d3c779f7a94c2a06991233b548ce29680bde1a9765bdbe212255d1bb2882be67.png

Interpolation is a kind of data driven model for developing a mathematical model from data that can be used to predict new values. These models are not based on physics, but they can be used for predicting new values, estimating derivatives, integrals, etc. Of course, you must be careful with extrapolation; all polynomials tend to ± infinity eventually, which is probably not physically relevant in most cases.

@@ -817,9 +817,9 @@

Eigenvalues -
array([[1.85633723, 0.90775149, 1.58462077],
-       [0.90775149, 0.05403518, 0.78648743],
-       [1.58462077, 0.78648743, 0.24651104]])
+
array([[0.46302306, 0.49799434, 1.13597004],
+       [0.49799434, 0.14431942, 0.78988401],
+       [1.13597004, 0.78988401, 0.32121277]])
@@ -834,7 +834,7 @@

Eigenvalues -
array([1.8263237 , 0.27801759, 0.04618801])
+
array([3.75159824, 0.61051038, 0.00993245])
@@ -846,8 +846,8 @@

Eigenvalues -
array([ 1.30646303+0.j        , -0.13209265+0.16514484j,
-       -0.13209265-0.16514484j])
+
array([1.42273837+0.j        , 0.1698839 +0.23583954j,
+       0.1698839 -0.23583954j])
@@ -861,7 +861,7 @@

Eigenvalues -
(np.float64(2.150529299114675), np.float64(2.150529299114675))
+
(np.float64(4.372041080266236), np.float64(4.372041080266236))
@@ -875,7 +875,7 @@

Eigenvalues -
np.float64(2.1505292991146763)
+
np.float64(4.37204108026624)
@@ -889,7 +889,7 @@

Eigenvalues -
(np.float64(0.02345196576353182), np.float64(0.023451965763531778))
+
(np.float64(0.022749189157166692), np.float64(0.02274918915716699))
@@ -903,10 +903,10 @@

Eigenvalues -
(array([1.8263237 , 0.27801759, 0.04618801]),
- array([[ 0.74278495,  0.65593421,  0.13424169],
-        [ 0.25337007, -0.08979379, -0.96319296],
-        [ 0.61973714, -0.74945806,  0.23289157]]))
+
(array([3.75159824, 0.61051038, 0.00993245]),
+ array([[-0.6396746 , -0.54985867,  0.53709575],
+        [-0.69886885,  0.12515565, -0.70421473],
+        [-0.31999801,  0.82582777,  0.46433799]]))
@@ -920,7 +920,7 @@

Eigenvalues -
(np.float64(1.0), np.float64(1.0))
+
(np.float64(1.0), np.float64(0.9999999999999999))
@@ -932,8 +932,8 @@

Eigenvalues -
(array([1.14342594, 0.38480519, 0.83306827]),
- array([1.35656576, 1.19794819, 0.24516878]))
+
(array([-1.7340062 , -1.57628126, -0.37860883]),
+ array([-2.39980212, -2.06284882,  2.01496747]))
@@ -946,8 +946,8 @@

Eigenvalues -
(array([1.35656576, 0.46273576, 1.13184063]),
- array([1.35656576, 0.46273576, 1.13184063]))
+
(array([-2.39980212, -2.62187516, -1.20050397]),
+ array([-2.39980212, -2.62187516, -1.20050397]))
@@ -959,7 +959,7 @@

Eigenvalues -
[np.float64(1.0), np.float64(1.0), np.float64(1.0)]
+
[np.float64(0.9999999999999999), np.float64(1.0), np.float64(1.0)]
@@ -1006,10 +1006,10 @@

Eigenvalues -
(array([0.04618801, 0.27801759, 1.8263237 ]),
- array([[ 0.13424169,  0.65593421,  0.74278495],
-        [-0.96319296, -0.08979379,  0.25337007],
-        [ 0.23289157, -0.74945806,  0.61973714]]))
+
(array([0.00993245, 0.61051038, 3.75159824]),
+ array([[ 0.53709575, -0.54985867, -0.6396746 ],
+        [-0.70421473,  0.12515565, -0.69886885],
+        [ 0.46433799,  0.82582777, -0.31999801]]))
diff --git a/book/18-linear-regression.html b/book/18-linear-regression.html index eb4fd6b..e950097 100644 --- a/book/18-linear-regression.html +++ b/book/18-linear-regression.html @@ -689,7 +689,7 @@

An example of polynomial fitting -../_images/b747ec2371f39cd1f7d71945b4c8817f4ca270bf117a0c1537ae5646e6c7d0f5.png +../_images/4f3dfbeed1f41acf96a7b0feadb4c261da2e64b56e4bd4c413110f94ff9a452f.png

@@ -705,7 +705,7 @@

An example of polynomial fitting -../_images/9d12ea0cd33411cf8b0e9d005b8bbb33490ed69b1c2bc382358ad0279d1d01f3.png +../_images/4b6b361e3fc6ad5b3c9baa110863b392ff8d0aaf943fdf6f30fc24f562fe5ed8.png

We previously claimed that solving this equation was equivalent to minimizing the summed squared errors. Here we demonstrate that is consistent with our observation for the first parameter.

@@ -731,7 +731,7 @@

An example of polynomial fitting -../_images/ba825d93fb2a27bd0db37046c4a3e555740877d8eafa6c5bb490c5d26385be75.png +../_images/4da5ed1b50c42b2132cc6118991d1c0770ca4150f969aee242e15abe0c395a20.png

@@ -885,7 +885,7 @@

Confidence intervals on the parameters -../_images/2f209ed670994ea105811ea89e14c3eeb31fffea86935e48d90efc1581b816ea.png +../_images/60b006a6c045530e25c7c582fd97f66e982496a1ea0382ba55345c2d4196fb5a.png

It is almost certainly not reasonable for the concentration of A to start increasing again after about 350 time units.

@@ -909,7 +909,7 @@

Regularization -../_images/cfd7de36aad87308051379396faffb34c5cbb161a71a8e51cac152ca2215d121.png +../_images/8a92a292957660f5ed4ca722ebc80809087cc6140d7b880fe4b02e4918175056.png

Our goal is to fit a linear regression model to this data. We want to avoid underfitting and overfitting. If we just fit polynomials to this data, we find some undesirable behavior. Let’s look at fits up to a 12th order polynomials.

@@ -940,7 +940,7 @@

Regularization +../_images/acc025ca1998f5d9e9bad0f8e40965df33f41a2cdd7e429806e36101b8394212.png

The most undesirable behavior is that the coefficients grow large, which puts a lot of weight in places we might not want. This also leads to wiggles in the fit, which are probably not reasonable. The solution to this issue is called regularization, which means we add a penalty to our objective function that serves to reduce the magnitude of the parameters. There are several approaches to regularization. In ridge regression we add an L2 penalty to the parameters, i.e. the sum of the parameters squared. In LASSO regression we add an L1 penalty to the parameters, i.e. the sum of the absolute values of the parameters.

@@ -1041,7 +1041,7 @@

Ridge regression +../_images/c747a39447bb3539d2f3a89fafe541c907c7d6e90afd7c5c5440d7e5aded0811.png

One way people have evaluated a reasonable value of λ is to look at how the coefficients vary with λ using a ridge plot. In this plot, you look for a range that balances the large swings associated with regular unconstrained regression and the damping caused by large values of λ. Here a value of \(10^{-6} \le \lambda \le 10^{-8}\) would be considered reasonable.

@@ -1062,7 +1062,7 @@

Ridge regression -../_images/fef5a985ae42236d3832b313de2d33cb285079353f9e1e375377cc879f836535.png +../_images/d7c078fc195a31c269ddb4479f334dd1dc2d936a4b37c22c7571b0fd4f03d5e7.png @@ -1107,7 +1107,7 @@

LASSO regression +../_images/28892f23b3c6fe8ec06ed76e8d5a0794c3a440aaba63a4a2ccdfcc161bf5d048.png

Now, we can explore the effect of λ more thoroughly.

@@ -1134,17 +1134,18 @@

LASSO regression
 0.00032      0.09    -0.01    -0.00    -0.16    -0.62     0.80     2.16     0.79    -4.96    -0.51     1.66
+
 0.00032
 

 

-
    0.01     -0.19    -0.00    -0.00    -0.00     1.02     0.39     0.00    -0.00    -1.75    -2.08     1.85
+
      0.09    -0.01    -0.00    -0.16    -0.62     0.80     2.16     0.79    -4.96    -0.51     1.66
+    0.01     -0.19    -0.00    -0.00    -0.00     1.02     0.39     0.00    -0.00    -1.75    -2.08     1.85
     0.32      0.28     0.29     0.00     0.00    -0.00    -0.00     0.00    -0.00    -1.99    -0.00     0.60
 

 

 
      10     -0.00    -0.00    -0.00    -0.00    -0.00    -0.00     0.00     0.00    -0.75    -0.00    -0.00
 

 

-../_images/ebe41146cb768be67f96b25f955209039bd7cd2c1a019ca6b6832d91960eb7b8.png
+../_images/67558850cb402449ec2b7581b20b687f1282d993d1f6f696e6d1213c389aba76.png
 

 

 
You can see that by increasing λ we are making more and more of the parameters go to zero; in other words the functions they correspond to are not part of the model any longer. This is called sparsifying the model. It reduces over-fitting by reducing the model complexity. Finding the most suitable value for λ requires some sophisticated programming and analysis, and it is an important topic in machine learning and data science.

diff --git a/book/23-gp.html b/book/23-gp.html
index 7b9edb2..eabe3b7 100644
--- a/book/23-gp.html
+++ b/book/23-gp.html
@@ -635,7 +635,7 @@ 
GPR by example
-../_images/e843db86712a5cd7467efd07194f02e8a457c1dd6695ffa81d8b6e966874ea05.png
+../_images/58c21f7fbc23aaed47011722733c6f1fa50bcfe539de3ff23d4c1cd919b3d263.png

So, what we need is a convenient way to compute the covariance between the points we know, and the points we want to estimate. To keep things simple for now, we consider a small data set. We need to be able to compute the distance from each known \(x_i\) to each known \(x_j\). Numpy array broadcasting makes this simple. We expand each array, and then take the difference.

@@ -780,7 +780,7 @@

GPR by example -
array([0.76288962, 0.77274368])
+
array([0.75330375, 0.76306034])
@@ -796,7 +796,7 @@

GPR by example -../_images/e37ef57ffefac8d222c50b27692a6aac43924eb3254ad484af0a1b09497ba846.png +../_images/dca3c9de7d173a0766f963b7dc3ff98c50bbf0ec9d74578ed98599db9d946fa4.png

That is not bad, but clearly not great. With a \(\lambda=0.15\), only one data point is contributing to the estimate, the other points have only small contributions because they are far from points we are estimating. This is a feature of the assumption we made about the covariance with \(\lambda\). This means we do not have enough data to make a very good estimate. We can see this if we try this with a much more dense data set.

@@ -830,7 +830,7 @@

GPR by example
Known data step size is 0.05
-../_images/0a0a22a3d9b0d0506638efb8edc3d32bb142a3ad93ab67c8cd2ef9c34eeb8886.png +../_images/83a1577f46094e1cba3d5948beccd062b8cedcd557a75f7fe10d7242fbf98dcd.png

Now you can see that we do very well in estimating the values. The length scale here might even be considered too short, since it is evident we are fitting trends in the noise.

@@ -861,7 +861,7 @@

GPR by example
Known data step size is 0.05
-../_images/51ae61a5f6d997c737173373e8481de1ef1e5ecdbf8e75ab419ce1469688ec11.png +../_images/341bdc3b050f4ba68cad9ccf13ee4798c48e64ec9aa5d65c030a5d5597ccb396.png
@@ -925,7 +925,7 @@

Underfitting in GPR -../_images/5e1d7b8b3069c491900c3984fa498baffe6bd1decc855a6c4cb07b1c299cf436.png +../_images/5986c267c9d057358ff601629b7f2e019dfc5f8b9580e37e0ca02f6172531e88.png

@@ -950,7 +950,7 @@

Overfitting in GPR -../_images/421908b13c6f6b6cc925d67e947ee98e2f446549dd6ee8984e8e3052478cd354.png +../_images/f2f5fdf657bb3eebe4f3987474aa5b31f51896c57bfe4b43e767a68d9648864a.png @@ -991,10 +991,10 @@

Finding the hyperparameters in GPR -
array([1.        , 0.06918223])
+
array([1.        , 0.07604382])
 

 

-../_images/32101ff10602a2a3d748e375158711bcb5c157c06bd868fbb6a08c317a33950d.png
+../_images/0078c77c15f6ec9d5f130da1d7a598dc3557214b37dd3656e97c953f7d627647.png
@@ -1029,10 +1029,10 @@

Finding the hyperparameters in GPR -
array([1.        , 0.23103398, 0.19547582])
+
array([1.        , 0.21758186, 0.19907133])
 

 

-../_images/9ada4ef7c06fdef0585a291812c9f4157dced25507eb359673564b0891e4fee9.png
+../_images/9a04f96d0fae87a971bd5388e563137dd21b4a45c0e99905e940609de5a5b7e5.png

Note that we still see some wiggles in the fit, indicating some minor degree of overfitting with the optimal hyperparameters. That is happening because we fit to all the data, and do not use any to estimate how good our fits are. You can use train/test data splits for GPR for this purpose as well, but it is out of the scope of the lecture today.

@@ -1067,7 +1067,7 @@

An example with a linear kernel -../_images/f646fe50df6cbb7c13fc314d21eefa949693fa4e5c21747c1a8ba97a6412a511.png +../_images/bd7a58efc7f4c7a41923d8847bc31148aab659d2c09bf806929650b9ca742a44.png

As before, we setup a log likelihood function and maximize it to get estimates for the parameters.

@@ -1087,18 +1087,18 @@

An example with a linear kernel -
  message: Desired error not necessarily achieved due to precision loss.
-  success: False
-   status: 2
-      fun: 7.474153244366276
-        x: [-2.913e-05  6.558e-01 -1.454e+00]
-      nit: 28
-      jac: [-2.861e-06  2.658e-05 -1.740e-05]
- hess_inv: [[ 2.627e-01 -9.650e-03 -1.080e-01]
-            [-9.650e-03  5.798e-02  1.002e-02]
-            [-1.080e-01  1.002e-02  4.506e-02]]
-     nfev: 383
-     njev: 93
+
  message: Optimization terminated successfully.
+  success: True
+   status: 0
+      fun: 7.583403798356311
+        x: [ 1.824e-06  6.524e-01 -1.460e+00]
+      nit: 36
+      jac: [ 2.384e-06 -6.676e-06  6.378e-06]
+ hess_inv: [[ 2.109e+00  1.193e-01  1.737e-01]
+            [ 1.193e-01  3.656e-02  1.994e-02]
+            [ 1.737e-01  1.994e-02  3.625e-02]]
+     nfev: 212
+     njev: 53
@@ -1126,7 +1126,7 @@

An example with a linear kernel -../_images/4852fbea862137755cac6c1f467fb00787f4a5ccd6ab360984abeeeffc8ea4d0.png +../_images/488bbe35b756d1325921acac7f2ecd70374cdb5cf05f8c2b0782f5c4d384c03b.png

Note that now, we get linear extrapolation, because we are using a linear kernel. Note also that the hyperparameters do not mean anything in particular to us. They do not include the slope or intercept. We can work those out pretty easily though. The intercept is just a prediction at \(x=0\):

@@ -1139,7 +1139,7 @@

An example with a linear kernel -
array([2.9661702])
+
array([2.96422963])
@@ -1152,7 +1152,7 @@

An example with a linear kernel -
np.float64(2.0404461859989325)
+
np.float64(2.029885996115033)
@@ -1186,7 +1186,7 @@

Uncertainty quantification in GPR -../_images/90258f9707b3a4a74619e5a8d92673813a2faef695fe751789c953e5648b68fe.png +../_images/eccff8c0b8f68c2c3f71123537e78b93889d1ea2e78057e85d26076d4200fb80.png

@@ -1204,7 +1204,7 @@

Combining kernels -../_images/0b6b782423938ab371b841bfd48d8e17b6d6b0df72f178e3ad92574cf7cc0d72.png +../_images/c46928c4c650ccb93a08674bd77b73b448f5447dbada401a1cbae5120091d83d.png

This looks like a sin wave superimposed on a line. A periodic kernel is defined as

@@ -1229,19 +1229,19 @@

Combining kernels
  message: Optimization terminated successfully.
   success: True
    status: 0
-      fun: 21.606438354036335
-        x: [ 6.859e-07  3.795e-01 -1.544e+00  6.539e-01  9.292e-01
-            -3.396e-01]
-      nit: 43
-      jac: [-2.384e-07 -1.669e-06  6.676e-06  1.431e-06 -8.821e-06
-             2.623e-06]
- hess_inv: [[ 8.108e-01  3.407e-03 ...  2.418e-03 -1.455e-03]
-            [ 3.407e-03  9.230e-03 ...  1.428e-03 -5.953e-04]
+      fun: 20.71345892592966
+        x: [ 8.658e-07  3.892e-01 -1.482e+00  6.442e-01  9.334e-01
+            -3.443e-01]
+      nit: 41
+      jac: [ 7.153e-06 -5.007e-06  1.907e-06  2.623e-06  9.537e-06
+             2.146e-06]
+ hess_inv: [[ 7.366e-01 -2.444e-02 ... -5.931e-03  7.700e-03]
+            [-2.444e-02  6.283e-03 ...  9.140e-04 -8.896e-04]
             ...
-            [ 2.418e-03  1.428e-03 ...  3.882e-04 -5.694e-05]
-            [-1.455e-03 -5.953e-04 ... -5.694e-05  7.279e-03]]
-     nfev: 476
-     njev: 68
+            [-5.931e-03  9.140e-04 ...  3.660e-04 -4.014e-05]
+            [ 7.700e-03 -8.896e-04 ... -4.014e-05  8.618e-03]]
+     nfev: 350
+     njev: 50
@@ -1268,7 +1268,7 @@

Combining kernels -../_images/7186048a57ed3bd29d0bef868bdd033ac059ca6f9e8a561693018abfe42fcc62.png +../_images/b94e6255186696b574671b7a264806404b8f1396efddd3e87008e20bb18acd88.png

Note that we get oscillatory + linear extrapolation behavior!

diff --git a/docs/about.html b/docs/about.html index 8e35b0b..f70758d 100644 --- a/docs/about.html +++ b/docs/about.html @@ -582,14 +582,14 @@

About pycse -
uname_result(system='Linux', node='fv-az1980-869', release='6.5.0-1022-azure', version='#23~22.04.1-Ubuntu SMP Thu May  9 17:59:24 UTC 2024', machine='x86_64')
+
uname_result(system='Linux', node='fv-az887-444', release='6.5.0-1022-azure', version='#23~22.04.1-Ubuntu SMP Thu May  9 17:59:24 UTC 2024', machine='x86_64')
 Linux
 ('64bit', 'ELF')
 x86_64
-fv-az1980-869
+fv-az887-444
 Linux-6.5.0-1022-azure-x86_64-with-glibc2.35
 x86_64
-('main', 'Jun 20 2024 16:02:53')
+('main', 'Jun 25 2024 18:25:01')
 3.11.9
 

 

@@ -613,10 +613,10 @@ 
About the Python packages
-
Python: 3.11.9 (main, Jun 20 2024, 16:02:53) [GCC 11.4.0]
+
Python: 3.11.9 (main, Jun 25 2024, 18:25:01) [GCC 11.4.0]
 numpy: 2.0.0
 scipy: 1.14.0
-matplotlib: 3.9.0
+matplotlib: 3.9.1
 

 

 

diff --git a/docs/beginner.html b/docs/beginner.html
index 0b1e260..315ea1d 100644
--- a/docs/beginner.html
+++ b/docs/beginner.html
@@ -776,7 +776,7 @@ 
A better fsolve
-
/tmp/ipykernel_2844/1685161423.py:2: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2862/1685161423.py:2: RuntimeWarning: The iteration is not making good progress, as measured by the 
   improvement from the last ten iterations.
   fsolve(f, 2.5)
 

diff --git a/docs/execution-statistics.html b/docs/execution-statistics.html
index 1bc8fcc..2b653f4 100644
--- a/docs/execution-statistics.html
+++ b/docs/execution-statistics.html
@@ -513,225 +513,225 @@ 
Build statistics
blog/basic-python

-
2024-06-27 15:19

+
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blog/data-analysis

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blog/math

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blog/nonlinear-algebra

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blog/plotting

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blog/programming

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blog/statistics

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blog/units

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book/13-constrained-optimization

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2024-06-27 15:20

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force

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2.17

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2.29

 
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book/15-intro-linear-algebra

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force

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101.95

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101.85

 
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book/16-linear-algebra

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2.19

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2.35

 
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book/17-linear-algebra-2

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2.39

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2.18

 
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book/18-linear-regression

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3.6

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book/20-autograd-applications

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2.87

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3.05

 
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book/21-machine-learning

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2.05

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2.26

 
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book/22-ml-2

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2.44

 
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book/23-gp

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2.78

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2.87

 
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docs/about

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2.21

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2.34

 
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docs/beginner

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2.1

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docs/utils

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diff --git a/searchindex.js b/searchindex.js
index 726c9c0..3607f54 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"alltitles": {"2-point vs. 4-point numerical derivatives": [[6, "point-vs-4-point-numerical-derivatives"]], "2d arrays": [[0, "d-arrays"]], "3D arrays": [[0, "id2"]], "A better fsolve": [[38, "a-better-fsolve"]], "A harder problem": [[3, "a-harder-problem"]], "A nonlinear BVP": [[2, "a-nonlinear-bvp"]], "A novel way to numerically estimate the derivative of a function - complex-step derivative approximation": [[6, "a-novel-way-to-numerically-estimate-the-derivative-of-a-function-complex-step-derivative-approximation"]], "A simple first order ode evaluated at specific points": [[2, "a-simple-first-order-ode-evaluated-at-specific-points"]], "A worked bvp problem": [[22, "a-worked-bvp-problem"]], "A worked example": [[20, "a-worked-example"]], "About pycse": [[37, "about-pycse"]], "About the Python packages": [[37, "about-the-python-packages"]], "About your python": [[10, "about-your-python"]], "Addition and subtraction": [[11, "addition-and-subtraction"], [27, "addition-and-subtraction"]], "Advanced function creation": [[0, "advanced-function-creation"]], "Advanced mathematical operators": [[0, "advanced-mathematical-operators"]], "Advanced selection of \u03bb": [[30, "advanced-selection-of"]], "Advanced string formatting": [[0, "advanced-string-formatting"]], "An application": [[26, "an-application"]], "An example of polynomial fitting": [[30, "an-example-of-polynomial-fitting"]], "An example with a linear kernel": [[34, "an-example-with-a-linear-kernel"]], "An example with curve_fit": [[24, "an-example-with-curve-fit"]], "Analytically solve a simple ODE": [[6, "analytically-solve-a-simple-ode"]], "Another approach to error propagation": [[11, "another-approach-to-error-propagation"]], "Another example": [[5, "another-example"], [5, "id2"]], "Another interpretation of neural networks": [[32, "another-interpretation-of-neural-networks"]], "Another way to parameterize an ODE - nested function": [[2, "another-way-to-parameterize-an-ode-nested-function"]], "Application in linear boundary value problems": [[28, "application-in-linear-boundary-value-problems"]], "Application in reaction engineering - Steady state CSTR": [[28, "application-in-reaction-engineering-steady-state-cstr"]], "Application to independent chemical reactions.": [[5, "application-to-independent-chemical-reactions"]], "Application to maximizing profit in a PFR": [[23, "application-to-maximizing-profit-in-a-pfr"]], "Application to roots of a polynomial": [[29, "application-to-roots-of-a-polynomial"]], "Applications": [[16, "applications"]], "Applications of lambda functions": [[0, "applications-of-lambda-functions"]], "Applications of linear algebra": [[28, "applications-of-linear-algebra"]], "Applications to optimization": [[29, "applications-to-optimization"]], "Are averages different": [[11, "are-averages-different"]], "Avoiding indexing in lists": [[38, "avoiding-indexing-in-lists"]], "Basic math": [[0, "basic-math"]], "Basic python usage": [[0, "basic-python-usage"]], "Basic statistics": [[11, "basic-statistics"]], "Better interpolate than never": [[3, "better-interpolate-than-never"]], "Boundary value equations": [[2, "boundary-value-equations"]], "Boundary value problem in heat conduction": [[2, "boundary-value-problem-in-heat-conduction"]], "Boundary value problems": [[22, "boundary-value-problems"]], "Brief comparison of GPR and NN": [[34, "brief-comparison-of-gpr-and-nn"]], "Brief intro to regular expressions": [[10, "brief-intro-to-regular-expressions"]], "Brief review": [[18, "brief-review"]], "Build statistics": [[39, "build-statistics"]], "CO": [[13, "co"]], "CO2": [[13, "co2"]], "Calculating a bubble point pressure of a mixture": [[13, "calculating-a-bubble-point-pressure-of-a-mixture"]], "Calling lapack directly from scipy": [[5, "calling-lapack-directly-from-scipy"]], "Changelog": [[40, "changelog"]], "Choice of activation functions in neural networks": [[33, "choice-of-activation-functions-in-neural-networks"]], "Combining kernels": [[34, "combining-kernels"]], "Combining numerical data with quad": [[6, "combining-numerical-data-with-quad"]], "Compare the equilibrium constants": [[13, "compare-the-equilibrium-constants"]], "Compare to a loop solution": [[10, "compare-to-a-loop-solution"]], "Compressibility variation from an implicit equation of state": [[31, "compressibility-variation-from-an-implicit-equation-of-state"]], "Compute areas": [[13, "compute-areas"]], "Compute gas phase pressures of each species": [[13, "compute-gas-phase-pressures-of-each-species"]], "Compute mole fractions and partial pressures": [[13, "compute-mole-fractions-and-partial-pressures"]], "Compute the t-score for our data": [[11, "compute-the-t-score-for-our-data"]], "Computing a pipe diameter": [[13, "computing-a-pipe-diameter"]], "Computing determinants from matrix decompositions": [[5, "computing-determinants-from-matrix-decompositions"]], "Computing equilibrium constants": [[13, "computing-equilibrium-constants"]], "Computing the pressure from a solid equation of state": [[31, "computing-the-pressure-from-a-solid-equation-of-state"]], "Concentration profile in a particle": [[22, "concentration-profile-in-a-particle"]], "Concluding remarks": [[35, "concluding-remarks"]], "Conclusions": [[10, "conclusions"]], "Confidence interval on an average": [[11, "confidence-interval-on-an-average"]], "Confidence intervals on the parameters": [[30, "confidence-intervals-on-the-parameters"]], "Conservation of mass in chemical reactions": [[13, "conservation-of-mass-in-chemical-reactions"]], "Constrained minimization": [[26, "constrained-minimization"]], "Constrained minimization to find equilibrium compositions": [[13, "constrained-minimization-to-find-equilibrium-compositions"]], "Constrained optimization": [[8, "constrained-optimization"], [26, "constrained-optimization"]], "Constrained optimization with Lagrange multipliers and autograd": [[31, "constrained-optimization-with-lagrange-multipliers-and-autograd"]], "Construct the Lagrange multiplier augmented function": [[8, "construct-the-lagrange-multiplier-augmented-function"]], "Constructing arrays": [[27, "constructing-arrays"]], "Controlling the format of printed variables": [[0, "controlling-the-format-of-printed-variables"]], "Counting roots": [[7, "counting-roots"]], "Coupled nonlinear equations": [[7, "coupled-nonlinear-equations"]], "Creating arrays in python": [[0, "creating-arrays-in-python"]], "Creating your own functions": [[0, "creating-your-own-functions"]], "Cubic equations of state": [[21, "cubic-equations-of-state"]], "Curve fitting to get overlapping peak areas": [[13, "curve-fitting-to-get-overlapping-peak-areas"]], "Customizing plots after the fact": [[9, "customizing-plots-after-the-fact"]], "Data analysis": [[1, "data-analysis"]], "Debugging/getting help": [[15, "debugging-getting-help"]], "Defining functions in python": [[0, "defining-functions-in-python"]], "Delay Differential Equations": [[2, "delay-differential-equations"]], "Derivatives by FFT": [[6, "derivatives-by-fft"]], "Derivatives by fitting a function and taking the analytical derivative": [[6, "derivatives-by-fitting-a-function-and-taking-the-analytical-derivative"]], "Derivatives by polynomial fitting": [[6, "derivatives-by-polynomial-fitting"]], "Derivatives of functions": [[20, "derivatives-of-functions"]], "Determining linear independence of a set of vectors": [[5, "determining-linear-independence-of-a-set-of-vectors"]], "Differential algebraic systems of equations": [[2, "differential-algebraic-systems-of-equations"]], "Differential equations": [[2, "differential-equations"]], "Diffusion": [[16, "diffusion"]], "Discussion": [[3, "discussion"]], "Division": [[11, "division"]], "Docker": [[41, "docker"]], "Documentation": [[40, "documentation"]], "Double-y axis plot": [[9, "double-y-axis-plot"]], "Effects of outliers on regression": [[25, "effects-of-outliers-on-regression"]], "Efficiency": [[13, "efficiency"]], "Eigenvalues": [[29, "eigenvalues"]], "Entropy-temperature chart": [[13, "entropy-temperature-chart"]], "Equality constraints": [[26, "equality-constraints"]], "Equilibrium constant based on mole numbers": [[13, "equilibrium-constant-based-on-mole-numbers"]], "Equilibrium constant calculation": [[13, "equilibrium-constant-calculation"]], "Equilibrium yield of WGS": [[13, "equilibrium-yield-of-wgs"]], "Error tolerance in numerical solutions to ODEs": [[2, "error-tolerance-in-numerical-solutions-to-odes"]], "Estimate the value of f at t=2.": [[3, "estimate-the-value-of-f-at-t-2"]], "Estimating the boiling point of water": [[13, "estimating-the-boiling-point-of-water"]], "Estimating the volume of a plug flow reactor": [[16, "estimating-the-volume-of-a-plug-flow-reactor"]], "Estimating the volume of a solid": [[16, "estimating-the-volume-of-a-solid"]], "Euler\u2019s method": [[17, "euler-s-method"]], "Evaluating line integrals": [[31, "evaluating-line-integrals"]], "Exponential and logarithmic functions": [[0, "exponential-and-logarithmic-functions"]], "Families of solutions to FODEs": [[18, "families-of-solutions-to-fodes"]], "Fancy, built-in colors in Python": [[9, "fancy-built-in-colors-in-python"]], "Find the derivative, and solve for where it is zero": [[23, "find-the-derivative-and-solve-for-where-it-is-zero"]], "Find the minimum distance from a point to a curve.": [[8, "find-the-minimum-distance-from-a-point-to-a-curve"]], "Find the volume of a PFR": [[16, "find-the-volume-of-a-pfr"]], "Finding equilibrium composition by direct minimization of Gibbs free energy on mole numbers": [[13, "finding-equilibrium-composition-by-direct-minimization-of-gibbs-free-energy-on-mole-numbers"]], "Finding equilibrium conversion": [[13, "finding-equilibrium-conversion"]], "Finding independent reactions": [[28, "finding-independent-reactions"]], "Finding maxima": [[23, "finding-maxima"]], "Finding the hyperparameters in GPR": [[34, "finding-the-hyperparameters-in-gpr"]], "Finding the maximum power of a photovoltaic device.": [[8, "finding-the-maximum-power-of-a-photovoltaic-device"]], "Finding the nth root of a periodic function": [[7, "finding-the-nth-root-of-a-periodic-function"]], "Finding the partial derivatives": [[8, "finding-the-partial-derivatives"]], "First guess": [[2, "first-guess"]], "First-order differential equations": [[17, "first-order-differential-equations"]], "Fit a line to numerical data": [[1, "fit-a-line-to-numerical-data"]], "Fitting a numerical ODE solution to data": [[1, "fitting-a-numerical-ode-solution-to-data"]], "Flexible nonlinear models for regression": [[32, "flexible-nonlinear-models-for-regression"]], "Float comparisons": [[42, "float-comparisons"]], "Fourth-order Runge-Kutta method": [[17, "fourth-order-runge-kutta-method"]], "Function extrema": [[23, "function-extrema"]], "Function integration by the Romberg method": [[6, 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"Graphical methods to help get initial guesses for multivariate nonlinear regression": [[1, "graphical-methods-to-help-get-initial-guesses-for-multivariate-nonlinear-regression"]], "H2O": [[13, "h2o"]], "Handling units with dimensionless equations": [[12, "handling-units-with-dimensionless-equations"]], "Handling units with the quantities module": [[12, "handling-units-with-the-quantities-module"]], "Headings and subheadings": [[15, "headings-and-subheadings"]], "Homogeneous, first-order linear differential equations": [[17, "homogeneous-first-order-linear-differential-equations"]], "Improved interpolation?": [[3, "improved-interpolation"]], "Indexing vectors and arrays in Python": [[0, "indexing-vectors-and-arrays-in-python"]], "Inequality constraints": [[26, "inequality-constraints"]], "Integrating a batch reactor design equation": [[13, "integrating-a-batch-reactor-design-equation"]], "Integrating equations in Python": [[6, "integrating-equations-in-python"]], "Integrating functions in python": [[6, "integrating-functions-in-python"]], "Integrating the batch reactor mole balance": [[13, "integrating-the-batch-reactor-mole-balance"]], "Integration in Python": [[16, "integration-in-python"]], "Interpolate on f(t) then invert the interpolated number": [[3, "interpolate-on-f-t-then-invert-the-interpolated-number"]], "Interpolating between data points": [[29, "interpolating-between-data-points"]], "Interpolation": [[3, "interpolation"], [29, "interpolation"]], "Interpolation libraries": [[29, "interpolation-libraries"]], "Interpolation of data": [[3, "interpolation-of-data"]], "Interpolation schemes": [[34, "interpolation-schemes"]], "Interpolation with splines": [[3, "interpolation-with-splines"]], "Interpretation": [[11, "interpretation"]], "Introduction to Python and Jupyter": [[14, "introduction-to-python-and-jupyter"]], "Introduction to automatic differentiation": [[31, "introduction-to-automatic-differentiation"]], "Introduction to linear algebra": [[27, 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"choice-of-activation-functions-in-neural-networks"]], "Combining kernels": [[34, "combining-kernels"]], "Combining numerical data with quad": [[6, "combining-numerical-data-with-quad"]], "Compare the equilibrium constants": [[13, "compare-the-equilibrium-constants"]], "Compare to a loop solution": [[10, "compare-to-a-loop-solution"]], "Compressibility variation from an implicit equation of state": [[31, "compressibility-variation-from-an-implicit-equation-of-state"]], "Compute areas": [[13, "compute-areas"]], "Compute gas phase pressures of each species": [[13, "compute-gas-phase-pressures-of-each-species"]], "Compute mole fractions and partial pressures": [[13, "compute-mole-fractions-and-partial-pressures"]], "Compute the t-score for our data": [[11, "compute-the-t-score-for-our-data"]], "Computing a pipe diameter": [[13, "computing-a-pipe-diameter"]], "Computing determinants from matrix decompositions": [[5, "computing-determinants-from-matrix-decompositions"]], "Computing 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"defining-functions-in-python"]], "Delay Differential Equations": [[2, "delay-differential-equations"]], "Derivatives by FFT": [[6, "derivatives-by-fft"]], "Derivatives by fitting a function and taking the analytical derivative": [[6, "derivatives-by-fitting-a-function-and-taking-the-analytical-derivative"]], "Derivatives by polynomial fitting": [[6, "derivatives-by-polynomial-fitting"]], "Derivatives of functions": [[20, "derivatives-of-functions"]], "Determining linear independence of a set of vectors": [[5, "determining-linear-independence-of-a-set-of-vectors"]], "Differential algebraic systems of equations": [[2, "differential-algebraic-systems-of-equations"]], "Differential equations": [[2, "differential-equations"]], "Diffusion": [[16, "diffusion"]], "Discussion": [[3, "discussion"]], "Division": [[11, "division"]], "Docker": [[41, "docker"]], "Documentation": [[40, "documentation"]], "Double-y axis plot": [[9, "double-y-axis-plot"]], "Effects of outliers on regression": [[25, 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"picasso-s-short-lived-blue-period-with-python"]], "Plane Poiseuille flow - BVP solve by shooting method": [[2, "plane-poiseuille-flow-bvp-solve-by-shooting-method"]], "Plane poiseuelle flow solved by finite difference": [[2, "plane-poiseuelle-flow-solved-by-finite-difference"]], "Plot customizations - Modifying line, text and figure properties": [[9, "plot-customizations-modifying-line-text-and-figure-properties"]], "Plot how the \\Delta G varies with temperature": [[13, "plot-how-the-delta-g-varies-with-temperature"]], "Plotting": [[9, "plotting"]], "Plotting ODE solutions in cylindrical coordinates": [[2, "plotting-ode-solutions-in-cylindrical-coordinates"]], "Plotting two datasets with very different scales": [[9, "plotting-two-datasets-with-very-different-scales"]], "Plug flow reactor with a pressure drop": [[13, "plug-flow-reactor-with-a-pressure-drop"]], "Polynomials in Python": [[21, "polynomials-in-python"]], "Polynomials in python": [[6, "polynomials-in-python"]], "Potential 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"Simpson method https://docs.scipy.org/doc/scipy-0.18.1/reference/generated/scipy.integrate.simps.html#scipy.integrate.simps": [[16, "simpson-method-https-docs-scipy-org-doc-scipy-0-18-1-reference-generated-scipy-integrate-simps-html-scipy-integrate-simps"]], "Simulating the events feature of Matlab\u2019s ode solvers": [[2, "simulating-the-events-feature-of-matlab-s-ode-solvers"]], "Smooth transitions between discontinuous functions": [[6, "smooth-transitions-between-discontinuous-functions"]], "Smooth transitions between two constants": [[6, "smooth-transitions-between-two-constants"]], "Solutions to first-order differential equations by integration": [[17, "solutions-to-first-order-differential-equations-by-integration"]], "Solve the quadratic equation": [[6, "solve-the-quadratic-equation"]], "Solving Bessel\u2019s Equation numerically": [[2, "solving-bessel-s-equation-numerically"]], "Solving CSTR design equations": [[13, "solving-cstr-design-equations"]], "Solving a parameterized 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