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zMR-s-g@xVo9Unzua1UXLp+@lSWE6lKyGmL&JPT;?tUe~8qwws zR;DRsX2uNh#+~-b+{lg$ALOm9Z*HpQ1LX%Df#I-~L4Fl_=zNgvDQ#qwF;y3f;0@@b zgACScsxps3&cYwb%Roy|7bZL)o9b#}1`Y$-34V%+oE^@-_;%0c`z?Tl3JwlDaBIz> z*aYR-@%aZ;qVTWRTq6F5jVOd4wZd>Ulo|=(f(+`NkYON%#D}Ub^672ls$$ZR>v;it zRDU=q3WpjFP_8CzL04c|&6(_3i~d;z?2-r{#eq<&et&<*0c5g3_B%YeJLqdKfpN0i z%bsGeowyQ<^xnXNlXe2NZ$Lambda Lambda Lambda

-
-
-
<function <lambda> at 0x7f57a8f0a2a0>
+
 
-
<function <lambda> at 0x7f57a8f0a200>
+
<function <lambda> at 0x7ff543fb6200>
 1
 3
 6
@@ -1326,7 +1326,7 @@ 

Creating arrays in python -
-
-
[[0.74187725 0.2789733  0.85562424]
- [0.44324658 0.8461447  0.70211027]
- [0.3480448  0.79650049 0.76008395]]
-[0.74187725 0.44324658 0.3480448 ]
-[0.22457315 0.30169902 0.43134817]
+
[[0.43615886 0.35305927 0.38172968]
+ [0.49819208 0.08390736 0.58993414]
+ [0.62070431 0.60793599 0.64336982]]
+[0.43615886 0.49819208 0.62070431]
+[0.5798703  0.81145878 0.02000972]
 
diff --git a/blog/differential-equations.html b/blog/differential-equations.html index 70e5ed3..672b1f3 100644 --- a/blog/differential-equations.html +++ b/blog/differential-equations.html @@ -1618,7 +1618,7 @@

Linear algebra approaches to solving systems of constant coefficient ODEs

-
/tmp/ipykernel_1978/2289625004.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
+
/tmp/ipykernel_1899/2289625004.py:4: DeprecationWarning: `row_stack` alias is deprecated. Use `np.vstack` directly.
   T = np.row_stack([t, t])
 
diff --git a/blog/interpolation.html b/blog/interpolation.html index 523b5ab..840166e 100644 --- a/blog/interpolation.html +++ b/blog/interpolation.html @@ -916,9 +916,9 @@

Interpolation with splines

- -
-
/tmp/ipykernel_2060/1343167527.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_1982/1343167527.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 +1363,7 @@

Combining numerical data with quad
/tmp/ipykernel_2060/1681875887.py:13: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_1982/1681875887.py:13: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   print(np.trapz(yfine, xfine))
 
@@ -1774,7 +1774,7 @@

The trapezoidal method of integration -
time elapsed = 0.0007147789001464844 sec
+
time elapsed = 0.0007112026214599609 sec
 1.9999983517708524
 
@@ -1795,7 +1795,7 @@

The trapezoidal method of integration -
time elapsed = 0.00023627281188964844 sec
+
time elapsed = 0.00022554397583007812 sec
 1.9999934070923728
 
@@ -1814,7 +1814,7 @@

The trapezoidal method of integration -
time elapsed = 7.200241088867188e-05 sec
+
time elapsed = 8.726119995117188e-05 sec
 1.999993407092373
 
@@ -2009,7 +2009,7 @@

Function integration by the Romberg method
/tmp/ipykernel_2060/1760157840.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_1982/1760157840.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))
 
@@ -2038,9 +2038,7 @@

Solve the quadratic equation
[(-b - sqrt(-4*a*c + b**2))/(2*a), (-b + sqrt(-4*a*c + b**2))/(2*a)]
-
-
-
-
/tmp/ipykernel_2090/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2010/4022852329.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_2090/4022852329.py:19: RuntimeWarning: The iteration is not making good progress, as measured by the 
+/tmp/ipykernel_2010/4022852329.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)
 
diff --git a/blog/plotting.html b/blog/plotting.html index 833c516..4b91c0e 100644 --- a/blog/plotting.html +++ b/blog/plotting.html @@ -1173,7 +1173,7 @@

Peak annotation in matplotlib -
@@ -1171,11 +1168,11 @@

About your python -
uname_result(system='Linux', node='fv-az735-304', 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-az1788-775', 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-az735-304
+fv-az1788-775
 Linux-6.5.0-1022-azure-x86_64-with-glibc2.35
 x86_64
 ('main', 'Jun 25 2024 18:25:01')
diff --git a/blog/statistics.html b/blog/statistics.html
index ef003e0..61c6767 100644
--- a/blog/statistics.html
+++ b/blog/statistics.html
@@ -1124,9 +1124,9 @@ 

Addition and subtraction -../_images/7cfffae1499a5c752f41c251276db302c0cf1a565087c4cc4aac4bd3bcbc9b3c.png -
0.5000410093954476
-0.5026134167129997
+../_images/d3bb7b9ec25aa1c3c907d125c9dea4082afb4610789e613af5a5e6f9a20769a5.png
+
0.5015934623528542
+0.5003768992300748
 0.5
 
@@ -1150,7 +1150,7 @@

Multiplication -
-
0.2918308673081258
+
0.2934053728479521
 0.2898598062432779
 
@@ -1194,7 +1194,7 @@

exponents -
1.7269169907396378
+
1.7263654373835455
 1.7236544062149992
 
@@ -1208,7 +1208,7 @@

exponents -
0.007449812297920145
+
0.007459637666347715
 0.007483647387490024
 
@@ -1236,7 +1236,7 @@

the chain rule in error propagation -
3.666447552884639
+
3.6726434783536566
 3.6180105030251086
 
@@ -1494,7 +1494,7 @@

Another approach to error propagation -
Ca(exit) = 0.005008757478474911+/-0.00017027719802033105
+
Ca(exit) = 0.005008860140460284+/-0.00017110482885036445
 
@@ -1526,8 +1526,8 @@

Random thoughts -
n = 0.7773698279605812
-You win!
+
n = 0.376772255942448
+you lose.
 
@@ -1551,10 +1551,10 @@

Random thoughts -
You won 5107 times (51.070000%)
+
You won 5048 times (50.480000%)
 
-../_images/f116823b864d6fc9064b3d8849aa6a6d2ed4f2b5e6f01587551b6b7cc7a0eca8.png +../_images/d5aac764cfc28098e69b26262424edbd71d99399e1576d9387eb2faa2af60ac9.png

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

@@ -1570,10 +1570,10 @@

Random thoughts -

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

@@ -1638,8 +1638,8 @@

Random thoughts -
-
/tmp/ipykernel_2236/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
+
/tmp/ipykernel_2158/2870202885.py:20: RuntimeWarning: divide by zero encountered in log
   diffg += R * T * x1 * np.log(x2 * P / P0)
-/tmp/ipykernel_2236/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
+/tmp/ipykernel_2158/2870202885.py:20: RuntimeWarning: invalid value encountered in multiply
   diffg += R * T * x1 * np.log(x2 * P / P0)
 
@@ -1576,9 +1576,9 @@

The Gibbs free energy of a reacting mixture and the equilibrium composition<
0.46959618248998036
 
-
-
/tmp/ipykernel_2236/4291268947.py:40: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2158/4291268947.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_2236/4291268947.py:41: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+/tmp/ipykernel_2158/4291268947.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))
 
@@ -3698,9 +3698,9 @@

What region is a point in -
/tmp/ipykernel_2236/339476128.py:64: RuntimeWarning: invalid value encountered in scalar divide
+
/tmp/ipykernel_2158/339476128.py:64: RuntimeWarning: invalid value encountered in scalar divide
   m1 = y1 / x1
-/tmp/ipykernel_2236/339476128.py:64: RuntimeWarning: divide by zero encountered in scalar divide
+/tmp/ipykernel_2158/339476128.py:64: RuntimeWarning: divide by zero encountered in scalar divide
   m1 = y1 / x1
 
diff --git a/book/00-intro.html b/book/00-intro.html index d58774f..58f3c21 100644 --- a/book/00-intro.html +++ b/book/00-intro.html @@ -816,7 +816,7 @@

scipy#< 906177281, 1325485569, 1443712780, 1963025921, -2063369727, 6098177, 553716061, 856765187, 588325653, 53813521, 16973318, -704760322, 1593690581, -470014715, 872205571, - 27328168, 50016, 32573, 228694960, 32573], + 27328168, 50016, 23789924, 0, 157549938], 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, 6.91198986e-310, 6.91198938e-310, - 6.91198918e-310, 6.91198986e-310, 6.91198938e-310, 6.91198918e-310, - 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, - 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, - 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, - 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, - 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, - 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, - 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, - 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, - 6.91198919e-310, 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, - 6.91198986e-310, 6.91198938e-310, 6.91198919e-310, 6.91198986e-310, - 6.91198938e-310, 6.91198919e-310]), + 'rlist': array([2.05718868e-001, 9.12099070e-001, 6.90578935e-310, 6.90578886e-310, + 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, + 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, + 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, + 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, + 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, + 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, + 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, + 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, + 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, + 6.90578867e-310, 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, + 6.90578935e-310, 6.90578886e-310, 6.90578867e-310, 6.90578935e-310, + 6.90578886e-310, 6.90578867e-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_2266/843280888.py:2: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2189/843280888.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/02-integration-1.html b/book/02-integration-1.html index 196f025..c2d52a1 100644 --- a/book/02-integration-1.html +++ b/book/02-integration-1.html @@ -727,7 +727,7 @@

numpy.trapz -
/tmp/ipykernel_2315/3604360318.py:4: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2236/3604360318.py:4: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
   np.trapz(y, x)
 
@@ -953,7 +953,7 @@

Estimating the volume of a plug flow reactor
/tmp/ipykernel_2315/4086341036.py:6: DeprecationWarning: `trapz` is deprecated. Use `trapezoid` instead, or one of the numerical integration functions in `scipy.integrate`.
+
/tmp/ipykernel_2236/4086341036.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])
 
@@ -990,7 +990,7 @@

Estimating the volume of a plug flow reactor
-
/tmp/ipykernel_2315/36719416.py:4: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.
+
 
-
<matplotlib.legend.Legend at 0x7f06edc26190>
+
<matplotlib.legend.Legend at 0x7f3d11b838d0>
 
../_images/0d992e05e05f92dc53b78d747da1fa725f1cb6fc66247847be3c6e338ba12bc2.png @@ -1114,7 +1114,7 @@

scipy.integrate.solve_ivp -
<matplotlib.legend.Legend at 0x7f06eb76a410>
+
<matplotlib.legend.Legend at 0x7f3d0f90f090>
 
../_images/0f3cb0f20de8a66563583063f2afeb51738e8b0db5b5ace82e7cb68a5289387e.png diff --git a/book/07-nla-1.html b/book/07-nla-1.html index 583b3ea..10d30dd 100644 --- a/book/07-nla-1.html +++ b/book/07-nla-1.html @@ -854,7 +854,7 @@

Derivatives of functions -
/tmp/ipykernel_2428/1845505028.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_2345/1845505028.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_2428/1563071673.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_2345/1563071673.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_2428/1563071673.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_2345/1563071673.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_2428/1619064557.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_2345/1619064557.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}')
 
@@ -1212,7 +1212,7 @@

Parameterized objective functions -
/tmp/ipykernel_2428/2335501430.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_2345/2335501430.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
 
diff --git a/book/08-nla-2.html b/book/08-nla-2.html index 2cdaa38..1a3e335 100644 --- a/book/08-nla-2.html +++ b/book/08-nla-2.html @@ -721,7 +721,7 @@

Cubic equations of state -
-
-
-
@@ -1104,12 +1095,7 @@

Systems of nonlinear equations -
Failed at guess [-9.998016728371766, -1.2858962573305965].
-The iteration is not making good progress, as measured by the 
-  improvement from the last five Jacobian evaluations.
-
-
-
-
-
/tmp/ipykernel_2510/2917585576.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_2427/2917585576.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_2535/3278399195.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_2452/3278399195.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)
 
diff --git a/book/15-intro-linear-algebra.html b/book/15-intro-linear-algebra.html index e3748b9..5557ec6 100644 --- a/book/15-intro-linear-algebra.html +++ b/book/15-intro-linear-algebra.html @@ -1257,7 +1257,7 @@

The determinant -

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/17-linear-algebra-2.html b/book/17-linear-algebra-2.html index 9551334..f43fee7 100644 --- a/book/17-linear-algebra-2.html +++ b/book/17-linear-algebra-2.html @@ -721,7 +721,7 @@

Interpolation libraries

-
/tmp/ipykernel_2687/3098855599.py:5: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2605/3098855599.py:5: RuntimeWarning: The iteration is not making good progress, as measured by the 
   improvement from the last ten iterations.
   fsolve(obj, 7)
 
@@ -817,9 +817,9 @@

Eigenvalues -
array([[0.76855485, 0.6428016 , 0.86663069],
-       [0.6428016 , 0.62741426, 0.58382858],
-       [0.86663069, 0.58382858, 1.51169623]])
+
array([[1.66449548, 1.4937402 , 1.58564427],
+       [1.4937402 , 1.63983854, 0.41286991],
+       [1.58564427, 0.41286991, 0.70594747]])
 
@@ -834,7 +834,7 @@

Eigenvalues -
array([3.48400685, 1.39901306, 0.27217613])
+
array([ 2.88850986e+00, -8.91770408e-01, -1.97512519e-04])
 
@@ -846,7 +846,8 @@

Eigenvalues -
array([ 1.43698492,  0.55271948, -0.17351181])
+
array([1.91481745+0.j       , 0.37018727+0.2031555j,
+       0.37018727-0.2031555j])
 
@@ -860,7 +861,7 @@

Eigenvalues -
(np.float64(5.155196038038639), np.float64(5.155196038038639))
+
(np.float64(1.9965419394516097), np.float64(1.9965419394516097))
 
@@ -874,7 +875,7 @@

Eigenvalues -
np.float64(5.155196038038636)
+
np.float64(1.9965419394516088)
 
@@ -888,7 +889,7 @@

Eigenvalues -
(np.float64(1.326633007746714), np.float64(1.3266330077467163))
+
(np.float64(0.0005087700530349978), np.float64(0.0005087700530348564))
 
@@ -902,10 +903,10 @@

Eigenvalues -
(array([3.48400685, 1.39901306, 0.27217613]),
- array([[-0.61009227, -0.61405389, -0.50072472],
-        [-0.41852523,  0.78635883, -0.45439676],
-        [-0.6727734 ,  0.06765803,  0.73674849]]))
+
(array([ 2.88850986e+00, -8.91770408e-01, -1.97512519e-04]),
+ array([[ 0.60631269,  0.78679919, -0.11546409],
+        [ 0.48216489, -0.47918907, -0.73341315],
+        [ 0.632378  , -0.38900498,  0.66990536]]))
 
@@ -919,7 +920,7 @@

Eigenvalues -
(np.float64(0.9999999999999997), np.float64(1.0))
+
(np.float64(0.9999999999999998), np.float64(0.9999999999999999))
 
@@ -931,8 +932,8 @@

Eigenvalues -
(array([-1.96077161, -1.59169991, -2.23466733]),
- array([-2.12556566, -2.13936795, -1.74452835]))
+
(array([1.07863149, 1.00048724, 1.28145979]),
+ array([ 1.7513402 ,  2.27267721, -0.33351916]))
 
@@ -945,8 +946,8 @@

Eigenvalues -
(array([-2.12556566, -1.45814476, -2.34394715]),
- array([-2.12556566, -1.45814476, -2.34394715]))
+
(array([1.7513402 , 1.39273803, 1.82663009]),
+ array([1.7513402 , 1.39273803, 1.82663009]))
 
@@ -958,7 +959,9 @@

Eigenvalues - -../_images/e8f54794e0ad99a7612ea969c08498a9574eb3241b176b5f28c7f5e473a090d4.png +../_images/9972b14218825004ee7646b9439d8cb3ab17760536508c561137dc9a17433ef6.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/845713084da04bc5b34af2192df7055bc2dca40f4c154906dde83365e70b418f.png +../_images/f959ec26bf2da087941b0fea4d2c44ef1c09777708518b31528c595c7ab23369.png

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

Overfitting in GPR -../_images/d58640c44c661fe4796f2189a8e69620ac1b47df108956199efc07ffba517233.png +../_images/27c30b74ef2ec755284c2c26261f5f273eec375a0528753468cbf2b90b227ec1.png

@@ -991,10 +991,10 @@

Finding the hyperparameters in GPR -
array([1.        , 0.07169676])
+
array([1.        , 0.06958857])
 
-../_images/e25645a6b4513e81cead526570cf50e0763e51bd4adbaa6e4f55ca3961a15502.png +../_images/055d22de8fe6bfa789188957b15f6cfd261a305f69b121d4170a5ead684bca45.png

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

@@ -1090,15 +1090,15 @@

An example with a linear kernel
  message: Optimization terminated successfully.
   success: True
    status: 0
-      fun: 7.419120639723772
-        x: [ 1.315e-05  6.555e-01 -1.439e+00]
-      nit: 27
-      jac: [ 3.755e-06  3.099e-06  2.265e-06]
- hess_inv: [[ 5.495e+00  4.856e-03  7.963e-03]
-            [ 4.856e-03  2.616e-02  1.512e-02]
-            [ 7.963e-03  1.512e-02  4.603e-02]]
-     nfev: 136
-     njev: 34
+      fun: 7.58903161523043
+        x: [-3.618e-05  6.409e-01 -1.497e+00]
+      nit: 30
+      jac: [-5.186e-06 -8.345e-06  6.795e-06]
+ hess_inv: [[ 3.542e+00 -5.724e-02 -4.603e-02]
+            [-5.724e-02  3.321e-02  2.011e-02]
+            [-4.603e-02  2.011e-02  5.654e-02]]
+     nfev: 148
+     njev: 37
 

@@ -1126,7 +1126,7 @@

An example with a linear kernel -../_images/1d9b62ecfa511f46e0554589fdd931fbf30e53e06d571b3bd19d10c086c2a392.png +../_images/f255c552287a09e48df546aa93cdb44f02351a30ec615678f145d99f069e9ade.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 - @@ -1204,7 +1204,7 @@

Combining kernels -../_images/ab65e772856dcb333573df4b4b8a711cc3ea679dc16b073225154f42a528f0a0.png +../_images/e52a56bd5dc6d476908251592114c05a3c7e5d3f208ff2e1e2707dcb1fa7c5ec.png

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

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

Combining kernels
  message: Desired error not necessarily achieved due to precision loss.
   success: False
    status: 2
-      fun: 21.045690685221263
-        x: [ 3.880e-07  4.178e-01 -1.354e+00  6.438e-01  9.417e-01
-            -3.355e-01]
-      nit: 46
-      jac: [ 4.768e-07  1.454e-05  1.121e-05 -1.001e-05  1.907e-06
-             3.576e-06]
- hess_inv: [[ 8.333e-01 -3.675e-02 ... -6.855e-03  7.983e-03]
-            [-3.675e-02  4.640e-03 ...  2.952e-04  4.366e-05]
+      fun: 22.29128263200922
+        x: [ 5.988e-06  4.474e-01 -1.286e+00  6.704e-01  2.340e+01
+             1.330e-02]
+      nit: 268
+      jac: [ 4.768e-06  7.677e-05 -2.193e-05  1.907e-06  2.623e-06
+             4.463e-04]
+ hess_inv: [[ 7.401e-01  5.643e-02 ... -8.424e-01  7.635e-04]
+            [ 5.643e-02  2.612e-02 ... -2.772e-01  2.630e-04]
             ...
-            [-6.855e-03  2.952e-04 ...  2.004e-04  2.517e-05]
-            [ 7.983e-03  4.366e-05 ...  2.517e-05  6.574e-03]]
-     nfev: 712
-     njev: 100
+            [-8.424e-01 -2.772e-01 ...  1.152e+02 -6.659e-02]
+            [ 7.635e-04  2.630e-04 ... -6.659e-02  4.852e-05]]
+     nfev: 2853
+     njev: 406
 

@@ -1268,7 +1268,7 @@

Combining kernels -../_images/8c9144b63e7177fb351d84a9eec6c9c75a13341f9d1ee7bec206dbd67bc04b74.png +../_images/b2708dc85b50097c1daab26de3599fafd5a1b1de3857fcdedb3a77dbf0f1c41a.png

Note that we get oscillatory + linear extrapolation behavior!

diff --git a/docs/about.html b/docs/about.html index 855f7d9..7f42e71 100644 --- a/docs/about.html +++ b/docs/about.html @@ -582,11 +582,11 @@

About pycse -
uname_result(system='Linux', node='fv-az735-304', 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-az1788-775', 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-az735-304
+fv-az1788-775
 Linux-6.5.0-1022-azure-x86_64-with-glibc2.35
 x86_64
 ('main', 'Jun 25 2024 18:25:01')
diff --git a/docs/beginner.html b/docs/beginner.html
index 4aaf3c8..cab9096 100644
--- a/docs/beginner.html
+++ b/docs/beginner.html
@@ -870,7 +870,7 @@ 

A better fsolve -
/tmp/ipykernel_2862/1685161423.py:2: RuntimeWarning: The iteration is not making good progress, as measured by the 
+
/tmp/ipykernel_2777/1685161423.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 de29b46..f392402 100644 --- a/docs/execution-statistics.html +++ b/docs/execution-statistics.html @@ -513,225 +513,225 @@

Build statistics

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-

2024-07-07 18:15

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blog/interpolation

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blog/linear-algebra

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blog/math

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blog/nonlinear-algebra

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blog/optimization

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blog/programming

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blog/statistics

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blog/units

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blog/worked-examples

-

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book/00-intro

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book/01-jupyter

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book/02-integration-1

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book/03-fode-1

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book/05-nth-odes

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book/07-nla-1

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book/08-nla-2

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docs/about

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docs/utils

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diff --git a/searchindex.js b/searchindex.js index 79a8037..a9ac5d5 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"]], "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, "function-integration-by-the-romberg-method"]], "Functions": [[15, "functions"]], "Functions on arrays of values": [[0, "functions-on-arrays-of-values"]], "Functions that return multiple values": [[15, "functions-that-return-multiple-values"]], "Functions with multiple arguments": [[15, "functions-with-multiple-arguments"]], "GPR Kernels": [[34, "gpr-kernels"]], "GPR by example": [[34, "gpr-by-example"]], "GPR libraries": [[34, "gpr-libraries"]], "Gaussian (radial basis function)": [[33, "gaussian-radial-basis-function"]], "Gaussian Process Regression": [[34, "gaussian-process-regression"]], "Gaussian process regression (GPR)": [[34, "gaussian-process-regression-gpr"]], "Getting a dictionary of counts": [[10, "getting-a-dictionary-of-counts"]], "Getting derivatives from implicit functions with autograd": [[31, "getting-derivatives-from-implicit-functions-with-autograd"]], "Gibbs energy minimization and the NIST webbook": [[13, "gibbs-energy-minimization-and-the-nist-webbook"]], "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, "introduction-to-linear-algebra"]], "Introduction to machine learning": [[32, "introduction-to-machine-learning"]], "Introduction to nonlinear algebra": [[20, "introduction-to-nonlinear-algebra"]], "Introduction to optimization": [[23, "introduction-to-optimization"]], "Introduction to solve_bvp": [[22, "introduction-to-solve-bvp"]], "Introduction to statistical data analysis": [[11, "introduction-to-statistical-data-analysis"]], "Invert f(t) then interpolate on 1/f": [[3, "invert-f-t-then-interpolate-on-1-f"]], "Is your ice cream float bigger than mine": [[6, "is-your-ice-cream-float-bigger-than-mine"]], "Isentropic compression of liquid to point 2": [[13, "isentropic-compression-of-liquid-to-point-2"]], "Isentropic expansion through turbine to point 4": [[13, "isentropic-expansion-through-turbine-to-point-4"]], "Isobaric heating to T3 in boiler where we make steam": [[13, "isobaric-heating-to-t3-in-boiler-where-we-make-steam"]], "Jupyter notebook introduction": [[14, "jupyter-notebook-introduction"]], "Keyboard shortcuts": [[15, "keyboard-shortcuts"]], "Know your tolerance": [[7, "know-your-tolerance"]], "LASSO regression": [[30, "lasso-regression"]], "Lambda Lambda Lambda": [[0, "lambda-lambda-lambda"]], "Last example": [[5, "last-example"]], "Lather, rinse and repeat": [[10, "lather-rinse-and-repeat"]], "Least Median regression": [[25, "least-median-regression"]], "Let fsolve do the work": [[2, "let-fsolve-do-the-work"]], "Leveraging linear algebra for iteration": [[28, "leveraging-linear-algebra-for-iteration"]], "Limitations of solutions by integration": [[17, "limitations-of-solutions-by-integration"]], "Linear algebra": [[5, "linear-algebra"]], "Linear algebra approaches to solving systems of constant coefficient ODEs": [[2, "linear-algebra-approaches-to-solving-systems-of-constant-coefficient-odes"]], "Linear algebra functions of arrays": [[27, "linear-algebra-functions-of-arrays"]], "Linear equality constraints for atomic mass conservation": [[13, "linear-equality-constraints-for-atomic-mass-conservation"]], "Linear least squares fitting with linear algebra": [[1, "linear-least-squares-fitting-with-linear-algebra"]], "Linear programming example with inequality constraints": [[8, "linear-programming-example-with-inequality-constraints"]], "Linear regression": [[30, "linear-regression"]], "Linear regression with confidence intervals (updated)": [[1, "linear-regression-with-confidence-intervals-updated"]], "Linear regression with confidence intervals.": [[1, "linear-regression-with-confidence-intervals"]], "Machine learning regression - flexible models with parameters": [[34, "machine-learning-regression-flexible-models-with-parameters"]], "Make two plots!": [[9, "make-two-plots"]], "Markdown": [[15, "markdown"]], "Math": [[6, "math"]], "Mathematical, scientific and engineering applications of autograd": [[31, "mathematical-scientific-and-engineering-applications-of-autograd"]], "Matrix algebra": [[27, "matrix-algebra"]], "Matrix algebra approach.": [[5, "matrix-algebra-approach"]], "Meet the steam tables": [[13, "meet-the-steam-tables"]], "Method 1": [[3, "method-1"], [7, "method-1"]], "Method 2": [[7, "method-2"]], "Method 2: switch the interpolation order": [[3, "method-2-switch-the-interpolation-order"]], "Method of continuity for nonlinear equation solving": [[7, "method-of-continuity-for-nonlinear-equation-solving"]], "Method of continuity for solving nonlinear equations - Part II": [[7, "method-of-continuity-for-solving-nonlinear-equations-part-ii"]], "Mimicking ode events in python": [[2, "mimicking-ode-events-in-python"]], "Minimal definition of a function with one input": [[15, "minimal-definition-of-a-function-with-one-input"]], "Minimizing the summed absolute errors": [[25, "minimizing-the-summed-absolute-errors"]], "Model selection": [[11, "model-selection"]], "Modeling a transient plug flow reactor": [[2, "modeling-a-transient-plug-flow-reactor"]], "Modern machine learning with neural networks": [[32, "modern-machine-learning-with-neural-networks"]], "More about using Jupyter notebooks": [[15, "more-about-using-jupyter-notebooks"]], "More user-friendly functions": [[38, "more-user-friendly-functions"]], "Multidimensional arrays": [[27, "multidimensional-arrays"]], "Multiple minima": [[23, "multiple-minima"]], "Multiplication": [[11, "multiplication"]], "Multiplication and division": [[27, "multiplication-and-division"]], "Nth order differential equations": [[19, "nth-order-differential-equations"]], "Near deficient rank": [[5, "near-deficient-rank"]], "Nested lists": [[10, "nested-lists"]], "Newton-Raphson method for finding solutions": [[20, "newton-raphson-method-for-finding-solutions"]], "Newton-Raphson method of minima finding": [[23, "newton-raphson-method-of-minima-finding"]], "Non-homogeneous linear first-order ODEs": [[17, "non-homogeneous-linear-first-order-odes"]], "Non-standard state \\Delta H and \\Delta G": [[13, "non-standard-state-delta-h-and-delta-g"]], "Nonlinear algebra": [[7, "nonlinear-algebra"], [20, "nonlinear-algebra"]], "Nonlinear curve fitting": [[1, "nonlinear-curve-fitting"]], "Nonlinear curve fitting by direct least squares minimization": [[1, "nonlinear-curve-fitting-by-direct-least-squares-minimization"]], "Nonlinear curve fitting with confidence intervals": [[1, "nonlinear-curve-fitting-with-confidence-intervals"]], "Nonlinear curve fitting with parameter confidence intervals": [[1, "nonlinear-curve-fitting-with-parameter-confidence-intervals"]], "Nonlinear regression": [[24, "nonlinear-regression"]], "Notable differences from Matlab": [[13, "notable-differences-from-matlab"]], "Now we solve for the zeros in the partial derivatives": [[8, "now-we-solve-for-the-zeros-in-the-partial-derivatives"]], "Numeric derivatives by differences": [[6, "numeric-derivatives-by-differences"]], "Numerical Simpsons rule": [[6, "numerical-simpsons-rule"]], "Numerical data integration": [[6, "numerical-data-integration"]], "Numerical integration of data": [[16, "numerical-integration-of-data"]], "Numerical propagation of errors": [[11, "numerical-propagation-of-errors"]], "Numerical quadrature - or integration of functions": [[16, "numerical-quadrature-or-integration-of-functions"]], "Numerical solution to a simple ode": [[2, "numerical-solution-to-a-simple-ode"]], "Numerical solutions to differential equations": [[17, "numerical-solutions-to-differential-equations"]], "Numerically calculating an effectiveness factor for a porous catalyst bead": [[13, "numerically-calculating-an-effectiveness-factor-for-a-porous-catalyst-bead"]], "ODEs with discontinuous forcing functions": [[2, "odes-with-discontinuous-forcing-functions"]], "Old-fashioned way with a loop": [[5, "old-fashioned-way-with-a-loop"]], "On the quad or trapz\u2019d in ChemE heaven": [[6, "on-the-quad-or-trapz-d-in-cheme-heaven"]], "Optimization": [[8, "optimization"]], "Ordinary differential equations": [[2, "ordinary-differential-equations"]], "Other pieces of a list": [[38, "other-pieces-of-a-list"]], "Other useful things to remember about polynomials": [[21, "other-useful-things-to-remember-about-polynomials"]], "Overfitting in GPR": [[34, "overfitting-in-gpr"]], "PYCSE": [[40, "module-pycse.PYCSE"]], "Parameter confidence intervals": [[24, "parameter-confidence-intervals"]], "Parameter estimation by directly minimizing summed squared errors": [[1, "parameter-estimation-by-directly-minimizing-summed-squared-errors"]], "Parameterized objective functions": [[20, "parameterized-objective-functions"]], "Partial differential equations": [[2, "partial-differential-equations"]], "Peak annotation in matplotlib": [[9, "peak-annotation-in-matplotlib"]], "Peak finding in Raman spectroscopy": [[13, "peak-finding-in-raman-spectroscopy"]], "Phase portraits of a system of ODEs": [[2, "phase-portraits-of-a-system-of-odes"]], "Picasso\u2019s short lived blue period with Python": [[9, "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 gotchas in linear algebra in numpy": [[5, "potential-gotchas-in-linear-algebra-in-numpy"]], "Predator-prey model example": [[18, "predator-prey-model-example"]], "Printing arrays": [[15, "printing-arrays"]], "Problem problems": [[20, "problem-problems"]], "Programming": [[10, "programming"]], "Python": [[14, "python"]], "Qualitative method for systems of ODEs": [[18, "qualitative-method-for-systems-of-odes"]], "Random thoughts": [[11, "random-thoughts"]], "Rank": [[27, "rank"]], "Read a Google Sheet into a pandas Dataframe": [[42, "read-a-google-sheet-into-a-pandas-dataframe"]], "Reading in delimited text files": [[1, "reading-in-delimited-text-files"]], "Reading parameter database text files in python": [[13, "reading-parameter-database-text-files-in-python"]], "Reduced row echelon form": [[5, "reduced-row-echelon-form"]], "Regression of data is a form of function 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