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a/_sources/orthogonality/least_squares.md b/_sources/orthogonality/least_squares.md index 4b07d49..c476f31 100644 --- a/_sources/orthogonality/least_squares.md +++ b/_sources/orthogonality/least_squares.md @@ -204,7 +204,7 @@ bests fits the data $(0,1),(1/4,3),(1/2,2),(3/4,-1),(1,0)$. ```{dropdown} Solution $$ -A = \left[ \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ 1 & 1 & 0 \end{array} \right] +A = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ 1 & 1 & 0 \end{array} \right] \hspace{10mm} \boldsymbol{y} = \left[ \begin{array}{r} 1 \\ 3 \\ 2 \\ -1 \\ 0 \end{array} \right] $$ diff --git a/_sources/systems/interpolation.md b/_sources/systems/interpolation.md index 130cebc..f34995e 100644 --- a/_sources/systems/interpolation.md +++ b/_sources/systems/interpolation.md @@ -377,7 +377,7 @@ Therefore $6a_3 + 2b_3 = 2b_4$ implies $b_4 = 0$, the equation $3a_3 + 2b_3 + c_ The value $t=2.5$ lies in the interval $[t_2,t_3]$ therefore we compute $$ -p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) - 2b_3 = -3 +p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) + 2b_3 = -3 $$ ```{image} /img/01_03_05.png diff --git a/_sources/systems/odes.md b/_sources/systems/odes.md index a621891..5131a29 100644 --- a/_sources/systems/odes.md +++ b/_sources/systems/odes.md @@ -414,7 +414,7 @@ $$ y'' = 2^t \ \ , \ \ y'(0) = 1 \ , \ \ y(1) = 0 $$ -using step size $h=0.25$. Use the forward difference formula and the boundary condition $y'(0)=0$ to approximate the boundary value $y_0$. +using step size $h=0.25$. Use the forward difference formula and the boundary condition $y'(0)=1$ to approximate the boundary value $y_0$. **Exercise 4.** Derive the general form of the linear system $A \boldsymbol{y} = \boldsymbol{b}$ for an equation with boundary conditions diff --git a/notebooks/01_linear_systems.html b/notebooks/01_linear_systems.html index 8513a35..5294bf7 100644 --- a/notebooks/01_linear_systems.html +++ b/notebooks/01_linear_systems.html @@ -647,9 +647,6 @@

Example: Resistor Network

-
- diff --git a/notebooks/03_polynomial_interpolation.html b/notebooks/03_polynomial_interpolation.html index d460034..2f1be61 100644 --- a/notebooks/03_polynomial_interpolation.html +++ b/notebooks/03_polynomial_interpolation.html @@ -565,7 +565,7 @@

Example 3 -
115575244.6289977
+
115575244.59700233
@@ -612,7 +612,7 @@

Example 4 -../_images/d2617a8cd6660d08b687e3d6e916943125722408d2d59dc7c17754ad0213d61a.png +../_images/2163db2a5cd31e0241dd5a7330d26275c78d6479665958b8210d2cd90ed63617.png

Yikes! The interpolating polynomial is very sensitive to small changes in the \(y\) values! That’s because the condition number is large!

diff --git a/notebooks/04_spline_interpolation.html b/notebooks/04_spline_interpolation.html index ee7c5fb..2479019 100644 --- a/notebooks/04_spline_interpolation.html +++ b/notebooks/04_spline_interpolation.html @@ -518,7 +518,7 @@

Example 1 -
14.530258040767446
+
14.530258040767464
@@ -573,7 +573,7 @@

Example 2 -../_images/21eb4f509f3b727e26d3294cd8fff95754db5b33c22e386509b51c01827dc09b.png +../_images/f923012dec899e9bd692ab0ca84d9b22c19e8a0fe0cb11fd4118840346cd8975.png

@@ -596,7 +596,7 @@

Example 3 -../_images/d626bc8f20d529cbdcebf5fc719e1121722e2867b111f9416301d294aab9971c.png +../_images/0660bd34e97594b966a583017de4d4e47608932b35972595a4f78b84ae652fcc.png

The cubic spline is not sensitive to small changes in the \(y\) values.

diff --git a/notebooks/06_least_squares_regression.html b/notebooks/06_least_squares_regression.html index bd6e275..ad6bb47 100644 --- a/notebooks/06_least_squares_regression.html +++ b/notebooks/06_least_squares_regression.html @@ -515,7 +515,7 @@

Example: Fake Noisy Linear Data -../_images/f2bd79cd55d4411f9fae6e4a734a1641613b71e06b9357f77bfb516cc855f9fa.png +../_images/c1d7963841af431efece1027ac45a40bf8432064bdc00b23298e4012fc752df3.png

Let’s use linear regression to retrieve the coefficients \(c_0\) and \(c_1\). Construct the matrix \(A\):

@@ -540,11 +540,11 @@

Example: Fake Noisy Linear Data -
array([[1.        , 0.58343377],
-       [1.        , 0.75391049],
-       [1.        , 0.80513638],
-       [1.        , 0.93628357],
-       [1.        , 0.85067217]])
+
array([[1.        , 0.10751451],
+       [1.        , 0.56699523],
+       [1.        , 0.92723053],
+       [1.        , 0.99217346],
+       [1.        , 0.39107857]])
@@ -558,7 +558,7 @@

Example: Fake Noisy Linear Data -
[ 6.95249672 -3.90471848]
+
[ 6.97163969 -3.98683101]
@@ -575,7 +575,7 @@

Example: Fake Noisy Linear Data -../_images/975c3a72088dd27d21d8d35eeef41e1343eb97c272bf30f1f121395bfd2dd511.png +../_images/7a1b77d531ba7fb9569773ace588d592561298f1207a20598f1b6fd86e670c40.png

@@ -643,7 +643,7 @@

Example: Fake Noisy Quadratic Data -../_images/b68652521a8b4da67d5745351536e402bdaed4a488e228f9c7f081726969cd65.png +../_images/082224710de71aa42f1c4e98ebca3b9f7d4cf059bf5760f2e943af9d172f60c4.png

Construct the matrix \(A\):

@@ -674,7 +674,7 @@

Example: Fake Noisy Quadratic Data -../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png +../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png

Let’s solve again but this time we use the QR decomposition:

@@ -699,7 +699,7 @@

Example: Fake Noisy Quadratic Data -../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png +../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png diff --git a/notebooks/07_pca.html b/notebooks/07_pca.html index 6c99e74..acd325c 100644 --- a/notebooks/07_pca.html +++ b/notebooks/07_pca.html @@ -447,7 +447,7 @@

2D Fake Data -../_images/a63ed999b02c23d40ac7eae8edcd0018b57cdad9462a912325329f3b1126aa84.png +../_images/dc01bdfb1b1966d9f89ed8881043cabb9ca50aa96d1b6da6726872f81855b4d3.png

Compute the SVD of the data matrix:

@@ -466,8 +466,8 @@

2D Fake Data -
array([[-0.86616362, -0.49976053],
-       [ 0.49976053, -0.86616362]])
+
array([[ 0.85839616,  0.51298736],
+       [-0.51298736,  0.85839616]])
@@ -482,7 +482,7 @@

2D Fake Data -../_images/f1af78ca31b876286c07bd3db5d26b6e32cb0dc695fdf7745d6f3693ee9df426.png +../_images/079acd8fa2efbe8c7d8fb5841d9163458cbbe0f78405e0b5362434a1d79461c3.png

diff --git a/notebooks/08_deblurring_images.html b/notebooks/08_deblurring_images.html index c517963..1e743e1 100644 --- a/notebooks/08_deblurring_images.html +++ b/notebooks/08_deblurring_images.html @@ -491,7 +491,7 @@

Blurring images by Toeplitz matrices -
24782.331042560716
+
24782.3310425676
@@ -596,7 +596,7 @@

Inverting the noise -../_images/d33c484752eb1c2fba9f92c4ab7cb2bc939e0132dc0c6150851813a063d38686.png +../_images/3d6e6ff5655379beb2128c93d707b85e9c1295301b43e6c1d0b0b72828a6edaa.png

@@ -616,7 +616,7 @@

Inverting the noise -../_images/62c7656f5e73e631ba350c817a6f3c27c8a4dd99578133c72d4fff7284a96800.png +../_images/d58f44359ceab84fbfaa8fb9ecbb7887424553d69ab11d422051f9b5dc4974e8.png

@@ -655,7 +655,7 @@

Truncated SVD -../_images/f27ddae0d8c2376fa811cfdc6f2fbfb96a774efa9a3c7c9b10798f0c50f070d0.png +../_images/83a6da2bf5ffef4035d455408f67cd2a9e8203172ce3777d4cf15bdfed38e4c3.png diff --git a/notebooks/09_computed_tomography.html b/notebooks/09_computed_tomography.html index 049141c..637c2e8 100644 --- a/notebooks/09_computed_tomography.html +++ b/notebooks/09_computed_tomography.html @@ -490,7 +490,7 @@

Measurement Matrix \(A\)
-../_images/13fdad9f7337c30e4e0744e9af0abdbd68f663282c9160434cd892e2322b800e.png +../_images/34e8fe21acab92a2a16ffadf8b3520fd2720dc48017194f6e7a73537b3f0b89c.png
diff --git a/notebooks/10_computing_eigenvalues.html b/notebooks/10_computing_eigenvalues.html index 038fbf5..f1594d8 100644 --- a/notebooks/10_computing_eigenvalues.html +++ b/notebooks/10_computing_eigenvalues.html @@ -638,8 +638,8 @@

Random matrices -
Power method:  384.70724040315235
-SciPy:  384.7072470931362
+
Power method:  399.7664965212194
+SciPy:  399.7664965222582
@@ -677,8 +677,8 @@

QR Iteration -
[[ 2.41421356e+00 -3.77213497e-16 -2.29396765e-16]
- [-1.45293347e-19  1.00000000e+00 -1.23531192e-16]
+
[[ 2.41421356e+00  2.38617379e-16  3.39797215e-16]
+ [ 1.45293347e-19  1.00000000e+00 -7.24650102e-17]
  [ 0.00000000e+00  1.45293347e-19 -4.14213562e-01]]
 

 

diff --git a/notebooks/12_fft.html b/notebooks/12_fft.html
index 6734ccc..996a50e 100644
--- a/notebooks/12_fft.html
+++ b/notebooks/12_fft.html
@@ -527,7 +527,7 @@ 
Filtering
-../_images/b7a9fec977b25f88e390483a45024a11b9b85cf93d5b8492bdcdf0c8887707a3.png
+../_images/4f07ece73dc0559a8a863f1726766e65569092195b260aae36a0db0fc2b9178e.png
@@ -550,7 +550,7 @@

Filtering -../_images/89b889f854358bfb37f8730168ce9e4d96675d7821ef79719a9e846cd79e7baa.png +../_images/9cd67069f4b2902cf3d3ebc67197c324ea232a189d52fef4bb0a39da1ffe7b4c.png

diff --git a/orthogonality/least_squares.html b/orthogonality/least_squares.html index a55649c..476d668 100644 --- a/orthogonality/least_squares.html +++ b/orthogonality/least_squares.html @@ -583,7 +583,7 @@

Exercises
\[\begin{split} -A = \left[ \begin{array}{rrr} 1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ 1 & 1 & 0 \end{array} \right] +A = \left[ \begin{array}{rrr} 1 & 1 & 0 \\ 1 & 0 & 1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \\ 1 & 1 & 0 \end{array} \right] \hspace{10mm} \boldsymbol{y} = \left[ \begin{array}{r} 1 \\ 3 \\ 2 \\ -1 \\ 0 \end{array} \right] \end{split}\]
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@@

Cubic Spline Interpolation\(t=2.5\) lies in the interval \([t_2,t_3]\) therefore we compute

\[ -p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) - 2b_3 = -3 +p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) + 2b_3 = -3 \]
../_images/01_03_05.png diff --git a/systems/odes.html b/systems/odes.html index 02a227e..6a2420d 100644 --- a/systems/odes.html +++ b/systems/odes.html @@ -736,7 +736,7 @@

Exercises\(h=0.25\). Use the forward difference formula and the boundary condition \(y'(0)=0\) to approximate the boundary value \(y_0\).

+

using step size \(h=0.25\). Use the forward difference formula and the boundary condition \(y'(0)=1\) to approximate the boundary value \(y_0\).

Exercise 4. Derive the general form of the linear system \(A \boldsymbol{y} = \boldsymbol{b}\) for an equation with boundary conditions

\[