deploy: e06a503

UBCMath · Oct 10, 2023 · d3a0c3a · d3a0c3a
1 parent e57778c
commit d3a0c3a
Show file tree

Hide file tree

Showing 46 changed files with 48 additions and 51 deletions.
diff --git a/_images/0660bd34e97594b966a583017de4d4e47608932b35972595a4f78b84ae652fcc.png b/_images/0660bd34e97594b966a583017de4d4e47608932b35972595a4f78b84ae652fcc.png
diff --git a/_images/079acd8fa2efbe8c7d8fb5841d9163458cbbe0f78405e0b5362434a1d79461c3.png b/_images/079acd8fa2efbe8c7d8fb5841d9163458cbbe0f78405e0b5362434a1d79461c3.png
diff --git a/_images/082224710de71aa42f1c4e98ebca3b9f7d4cf059bf5760f2e943af9d172f60c4.png b/_images/082224710de71aa42f1c4e98ebca3b9f7d4cf059bf5760f2e943af9d172f60c4.png
diff --git a/_images/13fdad9f7337c30e4e0744e9af0abdbd68f663282c9160434cd892e2322b800e.png b/_images/13fdad9f7337c30e4e0744e9af0abdbd68f663282c9160434cd892e2322b800e.png
diff --git a/_images/2163db2a5cd31e0241dd5a7330d26275c78d6479665958b8210d2cd90ed63617.png b/_images/2163db2a5cd31e0241dd5a7330d26275c78d6479665958b8210d2cd90ed63617.png
diff --git a/_images/21eb4f509f3b727e26d3294cd8fff95754db5b33c22e386509b51c01827dc09b.png b/_images/21eb4f509f3b727e26d3294cd8fff95754db5b33c22e386509b51c01827dc09b.png
diff --git a/_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png b/_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png
diff --git a/_images/34e8fe21acab92a2a16ffadf8b3520fd2720dc48017194f6e7a73537b3f0b89c.png b/_images/34e8fe21acab92a2a16ffadf8b3520fd2720dc48017194f6e7a73537b3f0b89c.png
diff --git a/_images/3d6e6ff5655379beb2128c93d707b85e9c1295301b43e6c1d0b0b72828a6edaa.png b/_images/3d6e6ff5655379beb2128c93d707b85e9c1295301b43e6c1d0b0b72828a6edaa.png
diff --git a/_images/4f07ece73dc0559a8a863f1726766e65569092195b260aae36a0db0fc2b9178e.png b/_images/4f07ece73dc0559a8a863f1726766e65569092195b260aae36a0db0fc2b9178e.png
diff --git a/_images/62c7656f5e73e631ba350c817a6f3c27c8a4dd99578133c72d4fff7284a96800.png b/_images/62c7656f5e73e631ba350c817a6f3c27c8a4dd99578133c72d4fff7284a96800.png
diff --git a/_images/7a1b77d531ba7fb9569773ace588d592561298f1207a20598f1b6fd86e670c40.png b/_images/7a1b77d531ba7fb9569773ace588d592561298f1207a20598f1b6fd86e670c40.png
diff --git a/_images/83a6da2bf5ffef4035d455408f67cd2a9e8203172ce3777d4cf15bdfed38e4c3.png b/_images/83a6da2bf5ffef4035d455408f67cd2a9e8203172ce3777d4cf15bdfed38e4c3.png
diff --git a/_images/89b889f854358bfb37f8730168ce9e4d96675d7821ef79719a9e846cd79e7baa.png b/_images/89b889f854358bfb37f8730168ce9e4d96675d7821ef79719a9e846cd79e7baa.png
diff --git a/_images/975c3a72088dd27d21d8d35eeef41e1343eb97c272bf30f1f121395bfd2dd511.png b/_images/975c3a72088dd27d21d8d35eeef41e1343eb97c272bf30f1f121395bfd2dd511.png
diff --git a/_images/9cd67069f4b2902cf3d3ebc67197c324ea232a189d52fef4bb0a39da1ffe7b4c.png b/_images/9cd67069f4b2902cf3d3ebc67197c324ea232a189d52fef4bb0a39da1ffe7b4c.png
diff --git a/_images/a63ed999b02c23d40ac7eae8edcd0018b57cdad9462a912325329f3b1126aa84.png b/_images/a63ed999b02c23d40ac7eae8edcd0018b57cdad9462a912325329f3b1126aa84.png
diff --git a/_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png b/_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png
diff --git a/_images/b68652521a8b4da67d5745351536e402bdaed4a488e228f9c7f081726969cd65.png b/_images/b68652521a8b4da67d5745351536e402bdaed4a488e228f9c7f081726969cd65.png
diff --git a/_images/b7a9fec977b25f88e390483a45024a11b9b85cf93d5b8492bdcdf0c8887707a3.png b/_images/b7a9fec977b25f88e390483a45024a11b9b85cf93d5b8492bdcdf0c8887707a3.png
diff --git a/_images/c1d7963841af431efece1027ac45a40bf8432064bdc00b23298e4012fc752df3.png b/_images/c1d7963841af431efece1027ac45a40bf8432064bdc00b23298e4012fc752df3.png
diff --git a/_images/d2617a8cd6660d08b687e3d6e916943125722408d2d59dc7c17754ad0213d61a.png b/_images/d2617a8cd6660d08b687e3d6e916943125722408d2d59dc7c17754ad0213d61a.png
diff --git a/_images/d33c484752eb1c2fba9f92c4ab7cb2bc939e0132dc0c6150851813a063d38686.png b/_images/d33c484752eb1c2fba9f92c4ab7cb2bc939e0132dc0c6150851813a063d38686.png
diff --git a/_images/d58f44359ceab84fbfaa8fb9ecbb7887424553d69ab11d422051f9b5dc4974e8.png b/_images/d58f44359ceab84fbfaa8fb9ecbb7887424553d69ab11d422051f9b5dc4974e8.png
diff --git a/_images/d626bc8f20d529cbdcebf5fc719e1121722e2867b111f9416301d294aab9971c.png b/_images/d626bc8f20d529cbdcebf5fc719e1121722e2867b111f9416301d294aab9971c.png
diff --git a/_images/dc01bdfb1b1966d9f89ed8881043cabb9ca50aa96d1b6da6726872f81855b4d3.png b/_images/dc01bdfb1b1966d9f89ed8881043cabb9ca50aa96d1b6da6726872f81855b4d3.png
diff --git a/_images/f1af78ca31b876286c07bd3db5d26b6e32cb0dc695fdf7745d6f3693ee9df426.png b/_images/f1af78ca31b876286c07bd3db5d26b6e32cb0dc695fdf7745d6f3693ee9df426.png
diff --git a/_images/f27ddae0d8c2376fa811cfdc6f2fbfb96a774efa9a3c7c9b10798f0c50f070d0.png b/_images/f27ddae0d8c2376fa811cfdc6f2fbfb96a774efa9a3c7c9b10798f0c50f070d0.png
diff --git a/_images/f2bd79cd55d4411f9fae6e4a734a1641613b71e06b9357f77bfb516cc855f9fa.png b/_images/f2bd79cd55d4411f9fae6e4a734a1641613b71e06b9357f77bfb516cc855f9fa.png
diff --git a/_images/f923012dec899e9bd692ab0ca84d9b22c19e8a0fe0cb11fd4118840346cd8975.png b/_images/f923012dec899e9bd692ab0ca84d9b22c19e8a0fe0cb11fd4118840346cd8975.png
diff --git 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
@@ -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
@@ -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
@@ -647,9 +647,6 @@ <h2>Example: Resistor Network<a class="headerlink" href="#example-resistor-netwo
  [ 5.]]
 </pre></div>
 </div>
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>
-</pre></div>
-</div>
 </div>
 </div>
 </section>

diff --git a/notebooks/03_polynomial_interpolation.html b/notebooks/03_polynomial_interpolation.html
@@ -565,7 +565,7 @@ <h2>Example 3<a class="headerlink" href="#example-3" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>115575244.6289977
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>115575244.59700233
 </pre></div>
 </div>
 </div>
@@ -612,7 +612,7 @@ <h2>Example 4<a class="headerlink" href="#example-4" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/d2617a8cd6660d08b687e3d6e916943125722408d2d59dc7c17754ad0213d61a.png" src="../_images/d2617a8cd6660d08b687e3d6e916943125722408d2d59dc7c17754ad0213d61a.png" />
+<img alt="../_images/2163db2a5cd31e0241dd5a7330d26275c78d6479665958b8210d2cd90ed63617.png" src="../_images/2163db2a5cd31e0241dd5a7330d26275c78d6479665958b8210d2cd90ed63617.png" />
 </div>
 </div>
 <p>Yikes! The interpolating polynomial is very sensitive to small changes in the <span class="math notranslate nohighlight">\(y\)</span> values! That’s because the condition number is large!</p>

diff --git a/notebooks/04_spline_interpolation.html b/notebooks/04_spline_interpolation.html
@@ -518,7 +518,7 @@ <h2>Example 1<a class="headerlink" href="#example-1" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>14.530258040767446
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>14.530258040767464
 </pre></div>
 </div>
 </div>
@@ -573,7 +573,7 @@ <h2>Example 2<a class="headerlink" href="#example-2" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/21eb4f509f3b727e26d3294cd8fff95754db5b33c22e386509b51c01827dc09b.png" src="../_images/21eb4f509f3b727e26d3294cd8fff95754db5b33c22e386509b51c01827dc09b.png" />
+<img alt="../_images/f923012dec899e9bd692ab0ca84d9b22c19e8a0fe0cb11fd4118840346cd8975.png" src="../_images/f923012dec899e9bd692ab0ca84d9b22c19e8a0fe0cb11fd4118840346cd8975.png" />
 </div>
 </div>
 </section>
@@ -596,7 +596,7 @@ <h2>Example 3<a class="headerlink" href="#example-3" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/d626bc8f20d529cbdcebf5fc719e1121722e2867b111f9416301d294aab9971c.png" src="../_images/d626bc8f20d529cbdcebf5fc719e1121722e2867b111f9416301d294aab9971c.png" />
+<img alt="../_images/0660bd34e97594b966a583017de4d4e47608932b35972595a4f78b84ae652fcc.png" src="../_images/0660bd34e97594b966a583017de4d4e47608932b35972595a4f78b84ae652fcc.png" />
 </div>
 </div>
 <p>The cubic spline is not sensitive to small changes in the <span class="math notranslate nohighlight">\(y\)</span> values.</p>

diff --git a/notebooks/06_least_squares_regression.html b/notebooks/06_least_squares_regression.html
@@ -515,7 +515,7 @@ <h3>Example: Fake Noisy Linear Data<a class="headerlink" href="#example-fake-noi
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/f2bd79cd55d4411f9fae6e4a734a1641613b71e06b9357f77bfb516cc855f9fa.png" src="../_images/f2bd79cd55d4411f9fae6e4a734a1641613b71e06b9357f77bfb516cc855f9fa.png" />
+<img alt="../_images/c1d7963841af431efece1027ac45a40bf8432064bdc00b23298e4012fc752df3.png" src="../_images/c1d7963841af431efece1027ac45a40bf8432064bdc00b23298e4012fc752df3.png" />
 </div>
 </div>
 <p>Let’s use linear regression to retrieve the coefficients <span class="math notranslate nohighlight">\(c_0\)</span> and <span class="math notranslate nohighlight">\(c_1\)</span>. Construct the matrix <span class="math notranslate nohighlight">\(A\)</span>:</p>
@@ -540,11 +540,11 @@ <h3>Example: Fake Noisy Linear Data<a class="headerlink" href="#example-fake-noi
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[1.        , 0.58343377],
-       [1.        , 0.75391049],
-       [1.        , 0.80513638],
-       [1.        , 0.93628357],
-       [1.        , 0.85067217]])
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[1.        , 0.10751451],
+       [1.        , 0.56699523],
+       [1.        , 0.92723053],
+       [1.        , 0.99217346],
+       [1.        , 0.39107857]])
 </pre></div>
 </div>
 </div>
@@ -558,7 +558,7 @@ <h3>Example: Fake Noisy Linear Data<a class="headerlink" href="#example-fake-noi
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 6.95249672 -3.90471848]
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 6.97163969 -3.98683101]
 </pre></div>
 </div>
 </div>
@@ -575,7 +575,7 @@ <h3>Example: Fake Noisy Linear Data<a class="headerlink" href="#example-fake-noi
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/975c3a72088dd27d21d8d35eeef41e1343eb97c272bf30f1f121395bfd2dd511.png" src="../_images/975c3a72088dd27d21d8d35eeef41e1343eb97c272bf30f1f121395bfd2dd511.png" />
+<img alt="../_images/7a1b77d531ba7fb9569773ace588d592561298f1207a20598f1b6fd86e670c40.png" src="../_images/7a1b77d531ba7fb9569773ace588d592561298f1207a20598f1b6fd86e670c40.png" />
 </div>
 </div>
 </section>
@@ -643,7 +643,7 @@ <h3>Example: Fake Noisy Quadratic Data<a class="headerlink" href="#example-fake-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/b68652521a8b4da67d5745351536e402bdaed4a488e228f9c7f081726969cd65.png" src="../_images/b68652521a8b4da67d5745351536e402bdaed4a488e228f9c7f081726969cd65.png" />
+<img alt="../_images/082224710de71aa42f1c4e98ebca3b9f7d4cf059bf5760f2e943af9d172f60c4.png" src="../_images/082224710de71aa42f1c4e98ebca3b9f7d4cf059bf5760f2e943af9d172f60c4.png" />
 </div>
 </div>
 <p>Construct the matrix <span class="math notranslate nohighlight">\(A\)</span>:</p>
@@ -674,7 +674,7 @@ <h3>Example: Fake Noisy Quadratic Data<a class="headerlink" href="#example-fake-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png" src="../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png" />
+<img alt="../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png" src="../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png" />
 </div>
 </div>
 <p>Let’s solve again but this time we use the QR decomposition:</p>
@@ -699,7 +699,7 @@ <h3>Example: Fake Noisy Quadratic Data<a class="headerlink" href="#example-fake-
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png" src="../_images/ad0a50622eb742a99bf52ab0594ef0c59e9efc146361c0ee76191b085f0d1ad5.png" />
+<img alt="../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png" src="../_images/2f02587282a758c5e0133e9de68c50658d542b94f7ef1ccb2eb9290a1091ef85.png" />
 </div>
 </div>
 </section>

diff --git a/notebooks/07_pca.html b/notebooks/07_pca.html
@@ -447,7 +447,7 @@ <h2>2D Fake Data<a class="headerlink" href="#d-fake-data" title="Permalink to th
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/a63ed999b02c23d40ac7eae8edcd0018b57cdad9462a912325329f3b1126aa84.png" src="../_images/a63ed999b02c23d40ac7eae8edcd0018b57cdad9462a912325329f3b1126aa84.png" />
+<img alt="../_images/dc01bdfb1b1966d9f89ed8881043cabb9ca50aa96d1b6da6726872f81855b4d3.png" src="../_images/dc01bdfb1b1966d9f89ed8881043cabb9ca50aa96d1b6da6726872f81855b4d3.png" />
 </div>
 </div>
 <p>Compute the SVD of the data matrix:</p>
@@ -466,8 +466,8 @@ <h2>2D Fake Data<a class="headerlink" href="#d-fake-data" title="Permalink to th
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[-0.86616362, -0.49976053],
-       [ 0.49976053, -0.86616362]])
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([[ 0.85839616,  0.51298736],
+       [-0.51298736,  0.85839616]])
 </pre></div>
 </div>
 </div>
@@ -482,7 +482,7 @@ <h2>2D Fake Data<a class="headerlink" href="#d-fake-data" title="Permalink to th
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/f1af78ca31b876286c07bd3db5d26b6e32cb0dc695fdf7745d6f3693ee9df426.png" src="../_images/f1af78ca31b876286c07bd3db5d26b6e32cb0dc695fdf7745d6f3693ee9df426.png" />
+<img alt="../_images/079acd8fa2efbe8c7d8fb5841d9163458cbbe0f78405e0b5362434a1d79461c3.png" src="../_images/079acd8fa2efbe8c7d8fb5841d9163458cbbe0f78405e0b5362434a1d79461c3.png" />
 </div>
 </div>
 </section>

diff --git a/notebooks/08_deblurring_images.html b/notebooks/08_deblurring_images.html
@@ -491,7 +491,7 @@ <h2>Blurring images by Toeplitz matrices<a class="headerlink" href="#blurring-im
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>24782.331042560716
+<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>24782.3310425676
 </pre></div>
 </div>
 </div>
@@ -596,7 +596,7 @@ <h2>Inverting the noise<a class="headerlink" href="#inverting-the-noise" title="
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/d33c484752eb1c2fba9f92c4ab7cb2bc939e0132dc0c6150851813a063d38686.png" src="../_images/d33c484752eb1c2fba9f92c4ab7cb2bc939e0132dc0c6150851813a063d38686.png" />
+<img alt="../_images/3d6e6ff5655379beb2128c93d707b85e9c1295301b43e6c1d0b0b72828a6edaa.png" src="../_images/3d6e6ff5655379beb2128c93d707b85e9c1295301b43e6c1d0b0b72828a6edaa.png" />
 </div>
 </div>
 <div class="cell docutils container">
@@ -616,7 +616,7 @@ <h2>Inverting the noise<a class="headerlink" href="#inverting-the-noise" title="
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/62c7656f5e73e631ba350c817a6f3c27c8a4dd99578133c72d4fff7284a96800.png" src="../_images/62c7656f5e73e631ba350c817a6f3c27c8a4dd99578133c72d4fff7284a96800.png" />
+<img alt="../_images/d58f44359ceab84fbfaa8fb9ecbb7887424553d69ab11d422051f9b5dc4974e8.png" src="../_images/d58f44359ceab84fbfaa8fb9ecbb7887424553d69ab11d422051f9b5dc4974e8.png" />
 </div>
 </div>
 </section>
@@ -655,7 +655,7 @@ <h2>Truncated SVD<a class="headerlink" href="#truncated-svd" title="Permalink to
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/f27ddae0d8c2376fa811cfdc6f2fbfb96a774efa9a3c7c9b10798f0c50f070d0.png" src="../_images/f27ddae0d8c2376fa811cfdc6f2fbfb96a774efa9a3c7c9b10798f0c50f070d0.png" />
+<img alt="../_images/83a6da2bf5ffef4035d455408f67cd2a9e8203172ce3777d4cf15bdfed38e4c3.png" src="../_images/83a6da2bf5ffef4035d455408f67cd2a9e8203172ce3777d4cf15bdfed38e4c3.png" />
 </div>
 </div>
 </section>

diff --git a/notebooks/09_computed_tomography.html b/notebooks/09_computed_tomography.html
@@ -490,7 +490,7 @@ <h2>Measurement Matrix <span class="math notranslate nohighlight">\(A\)</span><a
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/13fdad9f7337c30e4e0744e9af0abdbd68f663282c9160434cd892e2322b800e.png" src="../_images/13fdad9f7337c30e4e0744e9af0abdbd68f663282c9160434cd892e2322b800e.png" />
+<img alt="../_images/34e8fe21acab92a2a16ffadf8b3520fd2720dc48017194f6e7a73537b3f0b89c.png" src="../_images/34e8fe21acab92a2a16ffadf8b3520fd2720dc48017194f6e7a73537b3f0b89c.png" />
 </div>
 </div>
 </section>

diff --git a/notebooks/10_computing_eigenvalues.html b/notebooks/10_computing_eigenvalues.html
@@ -638,8 +638,8 @@ <h2>Random matrices<a class="headerlink" href="#random-matrices" title="Permalin
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Power method:  384.70724040315235
-SciPy:  384.7072470931362
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Power method:  399.7664965212194
+SciPy:  399.7664965222582
 </pre></div>
 </div>
 </div>
@@ -677,44 +677,44 @@ <h2>QR Iteration<a class="headerlink" href="#qr-iteration" title="Permalink to t
 
 Ak =
 
-[[ 2.00000000e+00 -7.07106781e-01  2.02930727e-17]
- [-7.07106781e-01  1.00000000e+00  7.07106781e-01]
+[[ 2.00000000e+00  7.07106781e-01 -1.77302319e-16]
+ [ 7.07106781e-01  1.00000000e+00  7.07106781e-01]
  [ 0.00000000e+00  7.07106781e-01  0.00000000e+00]]
 </pre></div>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Iteration k=2
 
 Ak =
 
-[[ 2.33333333e+00 -3.33333333e-01 -3.20655176e-16]
- [-3.33333333e-01  1.00000000e+00  3.33333333e-01]
+[[ 2.33333333e+00  3.33333333e-01  3.76358907e-16]
+ [ 3.33333333e-01  1.00000000e+00  3.33333333e-01]
  [ 0.00000000e+00  3.33333333e-01 -3.33333333e-01]]
 </pre></div>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Iteration k=3
 
 Ak =
 
-[[ 2.40000000e+00 -1.41421356e-01  2.07336995e-16]
- [-1.41421356e-01  1.00000000e+00  1.41421356e-01]
+[[ 2.40000000e+00  1.41421356e-01 -3.23313981e-16]
+ [ 1.41421356e-01  1.00000000e+00  1.41421356e-01]
  [ 0.00000000e+00  1.41421356e-01 -4.00000000e-01]]
 </pre></div>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Iteration k=4
 
 Ak =
 
-[[ 2.41176471e+00 -5.88235294e-02 -2.50505098e-16]
- [-5.88235294e-02  1.00000000e+00  5.88235294e-02]
+[[ 2.41176471e+00  5.88235294e-02  3.52409613e-16]
+ [ 5.88235294e-02  1.00000000e+00  5.88235294e-02]
  [ 0.00000000e+00  5.88235294e-02 -4.11764706e-01]]
 </pre></div>
 </div>
 <div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Iteration k=5
 
 Ak =
 
-[[ 2.41379310e+00 -2.43829925e-02  2.24955710e-16]
- [-2.43829925e-02  1.00000000e+00  2.43829925e-02]
+[[ 2.41379310e+00  2.43829925e-02 -3.36886114e-16]
+ [ 2.43829925e-02  1.00000000e+00  2.43829925e-02]
  [ 0.00000000e+00  2.43829925e-02 -4.13793103e-01]]
 </pre></div>
 </div>
@@ -736,8 +736,8 @@ <h2>QR Iteration<a class="headerlink" href="#qr-iteration" title="Permalink to t
 </div>
 </div>
 <div class="cell_output docutils container">
-<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 2.41421356e+00 -3.77213497e-16 -2.29396765e-16]
- [-1.45293347e-19  1.00000000e+00 -1.23531192e-16]
+<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[ 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]]
 </pre></div>
 </div>

diff --git a/notebooks/12_fft.html b/notebooks/12_fft.html
@@ -527,7 +527,7 @@ <h2>Filtering<a class="headerlink" href="#filtering" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/b7a9fec977b25f88e390483a45024a11b9b85cf93d5b8492bdcdf0c8887707a3.png" src="../_images/b7a9fec977b25f88e390483a45024a11b9b85cf93d5b8492bdcdf0c8887707a3.png" />
+<img alt="../_images/4f07ece73dc0559a8a863f1726766e65569092195b260aae36a0db0fc2b9178e.png" src="../_images/4f07ece73dc0559a8a863f1726766e65569092195b260aae36a0db0fc2b9178e.png" />
 </div>
 </div>
 <div class="cell docutils container">
@@ -550,7 +550,7 @@ <h2>Filtering<a class="headerlink" href="#filtering" title="Permalink to this he
 </div>
 </div>
 <div class="cell_output docutils container">
-<img alt="../_images/89b889f854358bfb37f8730168ce9e4d96675d7821ef79719a9e846cd79e7baa.png" src="../_images/89b889f854358bfb37f8730168ce9e4d96675d7821ef79719a9e846cd79e7baa.png" />
+<img alt="../_images/9cd67069f4b2902cf3d3ebc67197c324ea232a189d52fef4bb0a39da1ffe7b4c.png" src="../_images/9cd67069f4b2902cf3d3ebc67197c324ea232a189d52fef4bb0a39da1ffe7b4c.png" />
 </div>
 </div>
 </section>

diff --git a/orthogonality/least_squares.html b/orthogonality/least_squares.html
@@ -583,7 +583,7 @@ <h2>Exercises<a class="headerlink" href="#exercises" title="Permalink to this he
 </summary><div class="sd-summary-content sd-card-body docutils">
 <div class="math notranslate nohighlight">
 \[\begin{split}
-A = \left[ \begin{array}{rrr} 1 &amp; 1 &amp; 1 \\ 1 &amp; 0 &amp; 1 \\ 1 &amp; -1 &amp; 0 \\ 1 &amp; 0 &amp; -1 \\ 1 &amp; 1 &amp; 0 \end{array} \right]
+A = \left[ \begin{array}{rrr} 1 &amp; 1 &amp; 0 \\ 1 &amp; 0 &amp; 1 \\ 1 &amp; -1 &amp; 0 \\ 1 &amp; 0 &amp; -1 \\ 1 &amp; 1 &amp; 0 \end{array} \right]
 \hspace{10mm}
 \boldsymbol{y} = \left[ \begin{array}{r} 1 \\ 3 \\ 2 \\ -1 \\ 0 \end{array} \right]
 \end{split}\]</div>

diff --git a/searchindex.js b/searchindex.js
diff --git a/systems/interpolation.html b/systems/interpolation.html
@@ -695,7 +695,7 @@ <h2>Cubic Spline Interpolation<a class="headerlink" href="#cubic-spline-interpol
 <p>The value <span class="math notranslate nohighlight">\(t=2.5\)</span> lies in the interval <span class="math notranslate nohighlight">\([t_2,t_3]\)</span> therefore we compute</p>
 <div class="math notranslate nohighlight">
 \[
-p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) - 2b_3 = -3
+p''(2.5) = p_3''(2.5) = 6a_3(2.5 - 2) + 2b_3 = -3
 \]</div>
 <a class="reference internal image-reference" href="../_images/01_03_05.png"><img alt="../_images/01_03_05.png" class="align-center" src="../_images/01_03_05.png" style="width: 500px;" /></a>
 </div>

diff --git a/systems/odes.html b/systems/odes.html
@@ -736,7 +736,7 @@ <h2>Exercises<a class="headerlink" href="#exercises" title="Permalink to this he
 \[
 y'' = 2^t \ \ , \ \ y'(0) = 1 \ , \ \ y(1) = 0
 \]</div>
-<p>using step size <span class="math notranslate nohighlight">\(h=0.25\)</span>. Use the forward difference formula and the boundary condition <span class="math notranslate nohighlight">\(y'(0)=0\)</span> to approximate the boundary value <span class="math notranslate nohighlight">\(y_0\)</span>.</p>
+<p>using step size <span class="math notranslate nohighlight">\(h=0.25\)</span>. Use the forward difference formula and the boundary condition <span class="math notranslate nohighlight">\(y'(0)=1\)</span> to approximate the boundary value <span class="math notranslate nohighlight">\(y_0\)</span>.</p>
 <p><strong>Exercise 4.</strong> Derive the general form of the linear system <span class="math notranslate nohighlight">\(A \boldsymbol{y} = \boldsymbol{b}\)</span> for an equation with boundary conditions</p>
 <div class="math notranslate nohighlight">
 \[