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JSchoeberl committed Mar 6, 2024
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4 changes: 2 additions & 2 deletions _sources/iterative/Chebyshev.ipynb
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"source": [
"errhist = []\n",
"gfu.vec.data = ChebyIteration(a.mat, f.vec, pre, gamma1, gamma2, maxit=500,\n",
" # callback=lambda it,err: print(\"it=\",it,\"err=\",err) \n",
" callback = lambda it,err: errhist.append(err)\n",
" # callback=lambda it,err: print(\"it=\",it,\"err=\",err) \n",
" callback = lambda it,err: errhist.append(err)\n",
" )"
]
},
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10 changes: 5 additions & 5 deletions _sources/iterative/Richardson.ipynb
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"This means the new error is obtained from the old error by the error propagation matrix\n",
"\n",
"$$\n",
"e^{k+1} = (I - \\alpha A) \\, e^k\n",
"e^{k+1} = \\underbrace{(I - \\alpha A)}_{M} \\, e^k\n",
"$$\n",
"\n",
"Two strategies to verify convergence are:\n",
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"However, if $\\| I - \\alpha A \\| < 1$, then\n",
"\n",
"$$\n",
"\\| e^k \\| \\leq \\| I - \\alpha A \\| \\, \\| e^k \\|,\n",
"\\| e^{k+1} \\| \\leq \\| I - \\alpha A \\| \\, \\| e^k \\|,\n",
"$$\n",
"and the error decreases in every interation step. Note that the matrix norm is the operator norm generated by the vector norm.\n",
"proves that the error decreases in every iteration step. Note that the matrix norm is the operator norm generated by the vector norm.\n",
"\n",
"Some facts:\n",
"* If the norm $\\| \\cdot \\|$ is generated by an inner product $\\left< \\cdot, \\cdot \\right>$ (parallelogram identity), and $M$ is some self adjoint matrix with respect to this inner product, i.e.\n",
Expand All @@ -96,10 +96,10 @@
"\n",
"* If $\\left< \\cdot , \\cdot \\right>$ is the Euklidean inner product, then $M$ is self-adjoint exactly when $M$ is symmetric.\n",
" \n",
"* Every operator operator norm is bounded by the spectral radius. There exists some norm such that the operator norm is arbitrary close to the spectral radius, i.e.\n",
"* The spectral radius is bounded by every operator operator norm. There exists some norm such that the operator norm is arbitrary close to the spectral radius, i.e.\n",
" \n",
" $$\n",
" \\rho(M) = \\sup_{ \\text{norms} \\| \\cdot \\| } \\| M \\|\n",
" \\rho(M) = \\inf_{ \\text{norms} \\| \\cdot \\| } \\| M \\|\n",
" $$"
]
},
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40 changes: 7 additions & 33 deletions _sources/iterative/conjugategradients.ipynb
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29 changes: 5 additions & 24 deletions _sources/iterative/gradientmethod.ipynb
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20 changes: 19 additions & 1 deletion _sources/iterative/preconditioning.ipynb
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"$$\n"
]
},
{
"cell_type": "markdown",
"id": "88f9e839-3b46-4488-a3c8-a85bf4d8a74a",
"metadata": {},
"source": [
"**Exercise:**\n",
"Repeat the experiments with the forward-backward Gauss-Seidel preconditioner. You get it via\n",
"`pre = a.mat.CreateSmoother(GS=True)`. What do you observe for the spectral radius of $C^{-1} A$ ? "
]
},
{
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"id": "f9d7aeb3-fbd8-430d-806d-a4e5cdc89565",
"metadata": {},
"outputs": [],
"source": []
},
{
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"metadata": {},
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"source": []
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2,628 changes: 1,314 additions & 1,314 deletions iterative/Richardson.html
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29 changes: 16 additions & 13 deletions iterative/preconditioning.html
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Expand Down Expand Up @@ -590,25 +590,25 @@ <h2><span class="section-number">47.1. </span>The preconditioned Richardson iter
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.5628463859901822
1.3004844587339313
1.3642450612653387
1.4017278282832693
1.3004844587339315
1.3642450612653385
1.4017278282832695
1.428927912054942
1.4511979272962623
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1.489456390475471
1.451197927296262
1.4709697187344923
1.4894563904754712
1.5072557909082451
1.5246072026789994
1.5415208146951107
1.557860994429892
1.5578609944298918
1.5734139487420604
1.5879473334387257
1.601258905051369
1.6132090247120108
1.587947333438726
1.6012589050513686
1.613209024712011
1.6237346368041081
1.632846787976177
1.6328467879761772
1.6406169867162068
1.6471585575973842
1.6471585575973837
</pre></div>
</div>
</div>
Expand Down Expand Up @@ -876,6 +876,9 @@ <h2><span class="section-number">47.3. </span>Jacobi and Gauss Seidel Preconditi
\[
\left&lt; M_{FGS} x, M_{FGS} x \right&gt;_A = \left&lt; M_{BGS} M_{FGS} x, x \right&gt;_A \leq \| M_{FBGS} \|_A \| x \|_A^2
\]</div>
<p><strong>Exercise:</strong>
Repeat the experiments with the forward-backward Gauss-Seidel preconditioner. You get it via
<code class="docutils literal notranslate"><span class="pre">pre</span> <span class="pre">=</span> <span class="pre">a.mat.CreateSmoother(GS=True)</span></code>. What do you observe for the spectral radius of <span class="math notranslate nohighlight">\(C^{-1} A\)</span> ?</p>
</section>
</section>

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