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Commit 90ae0ca7 authored by Benjamin Berkels's avatar Benjamin Berkels
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improved the IPython notebook for Example 2.2.4

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%% Cell type:markdown id: tags: %% Cell type:markdown id: tags:
### IPython notebook for Example 2.2.4 from the lecture ### IPython notebook for Example 2.2.4 from the lecture
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
%matplotlib inline %matplotlib inline
import numpy as np import numpy as np
import matplotlib.pyplot as plt import matplotlib.pyplot as plt
from IPython.display import set_matplotlib_formats, display, Math from matplotlib_inline.backend_inline import set_matplotlib_formats
set_matplotlib_formats('svg') from IPython.display import display, Math
n = 40 set_matplotlib_formats("svg")
h = 1/n
display(Math(r'''A = h \begin{pmatrix} 1 & & & \\ 1 & 1 & &0 \\ \vdots &\vdots & \ddots & \\ 1 & 1& \dots & 1 \end{pmatrix}''')) n = 40
A = h*np.tril(np.ones((n, n), dtype=int), 0) h = 1 / n
display(Math(r'''y= \frac{1}{n}(1,1,\ldots,1)^T''')) display(Math(r"\text{Setting from Section 1.2:}"))
y = np.ones(n)/n display(Math(r"A = h \begin{pmatrix} 1 & & & \\ 1 & 1 & &0 \\ \vdots &\vdots & \ddots & \\ 1 & 1& \dots & 1 \end{pmatrix}\text{ (discrete integration matrix)}"))
display(Math(r'''\delta y = \frac{1}{n}(1,-1,1,-1, \ldots)^T''')) A = h * np.tril(np.ones((n, n), dtype=int), 0)
display(Math(r"y= \frac{1}{n}(1,1,\ldots,1)^T\text{ (low frequency input)}"))
y = np.ones(n) / n
display(Math(r"\delta y = \frac{1}{n}(1,-1,1,-1, \ldots)^T\text{ (high frequency pertubation)}"))
delta_y = y.copy() delta_y = y.copy()
delta_y[1::2] = -1/n delta_y[1::2] = -1 / n
display(Math(r'''\Rightarrow \|y\|_2= \|\delta y\|_2= \frac{1}{\sqrt{n}}''')) display(Math(r"\Rightarrow \|y\|_2= \|\delta y\|_2= \frac{1}{\sqrt{n}}"))
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
display(Math(r'U^TAV=\Sigma')) display(Math(r"U^TAV=\Sigma"))
U, sigma, VT = np.linalg.svd(A) U, sigma, VT = np.linalg.svd(A)
display(Math(r'\hat y=U^Ty')) display(Math(r"\hat y=U^Ty"))
y_hat = np.matmul(U.T, y) y_hat = np.matmul(U.T, y)
display(Math(r'\hat{\delta y}:= U^T \delta y')) display(Math(r"\hat{\delta y}:= U^T \delta y"))
delta_y_hat = np.matmul(U.T, delta_y) delta_y_hat = np.matmul(U.T, delta_y)
sigma_inv = 1/sigma sigma_inv = 1 / sigma
j = np.arange(1, 41) j = np.arange(1, 41)
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.yscale('log') plt.yscale("log")
plt.plot(j, np.abs(y_hat), '.', label=r'$|(U^Ty)_j|$') plt.plot(j, np.abs(y_hat), ".", label=r"$|(U^Ty)_j|$")
plt.plot(j, np.abs(delta_y_hat), '+', label=r'$|(U^T \delta y)_j|$') plt.plot(j, np.abs(delta_y_hat), "+", label=r"$|(U^T \delta y)_j|$")
plt.legend() plt.legend()
plt.show() plt.show()
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.close() plt.close()
plt.yscale('log') plt.yscale("log")
plt.plot(j, sigma_inv, '*', label=r'$\sigma_j^{-1}$') plt.plot(j, sigma_inv, "*", label=r"$\sigma_j^{-1}$")
plt.legend() plt.legend()
plt.show() plt.show()
``` ```
%% Cell type:code id: tags: %% Cell type:code id: tags:
``` python ``` python
plt.close() plt.close()
plt.yscale('log') plt.yscale("log")
plt.plot(j, sigma_inv*np.abs(y_hat), '.', label=r'$\sigma_j^{-1}|(U^Ty)_j|$') plt.plot(j, sigma_inv * np.abs(y_hat), ".", label=r"$\sigma_j^{-1}|(U^Ty)_j|$")
plt.plot(j, sigma_inv*np.abs(delta_y_hat), '+', label=r'$\sigma_j^{-1}|(U^T \delta y)_j|$') plt.plot(j, sigma_inv * np.abs(delta_y_hat), "+", label=r"$\sigma_j^{-1}|(U^T \delta y)_j|$")
plt.legend() plt.legend()
plt.show() plt.show()
display(Math(r'''\Rightarrow\ \|A^{-1} y\|_2 = \|\Sigma^{-1} \hat{y}\|_2 \ll \|A^{-1} \delta y\|_2= \|\Sigma^{-1} \hat{\delta y}\|_2\ \Rightarrow\ Q(y,\delta y) \gg 1''')) display(Math(r"\Rightarrow\ \|A^{-1} y\|_2 = \|\Sigma^{-1} \hat{y}\|_2 \ll \|A^{-1} \delta y\|_2= \|\Sigma^{-1} \hat{\delta y}\|_2\ \Rightarrow\ Q(y,\delta y) \gg 1"))
```
%% Cell type:code id: tags:
``` python
``` ```
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