Abstract
Iterative methods are used to approximate the solution of different kinds of linear or non-linear problems and also to construct curves (and surfaces).
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Notes
- 1.
Show that \(\varphi _{1}: I \rightarrow I\) and \(0 <\varphi _{1}(x) < 1\) for all x ∈ I.
- 2.
Show that \(x_{1} > \sqrt{2}\) whenever \(0 < x_{0} < \sqrt{2}\).
- 3.
For the loop, use while (iter<maxiter) & (error>precis)
- 4.
Answer:
$$\displaystyle{\boldsymbol{B}_{J} = \frac{1} {5}\left [\begin{array}{*{10}c} 0 & 2 & 2\\ 2 & 0 & 3 \\ 2 & 3 & 0 \end{array} \right ],\quad \boldsymbol{c}_{J}:= \frac{1} {5}\left [\begin{array}{*{10}c} 1\\ 0 \\ 0 \end{array} \right ].}$$ - 5.
Subtract \(\boldsymbol{z} = \boldsymbol{B}_{J}\boldsymbol{z} + \boldsymbol{c}_{J}\) from \(\boldsymbol{x}_{k+1} = \boldsymbol{B}_{J}\boldsymbol{x}_{k} + \boldsymbol{c}_{J}\).
- 6.
Compute \(\Vert \boldsymbol{B}_{GS}\Vert _{\infty }\).
- 7.
Use \(\boldsymbol{A} = \boldsymbol{D} -\boldsymbol{E} -\boldsymbol{F}\) to write \(\boldsymbol{B}_{J}\) and \(\boldsymbol{c}_{J}\) with the matrices \(\boldsymbol{D},\boldsymbol{E},\boldsymbol{F}\) and the vector \(\boldsymbol{b}\).
- 8.
Use diag, ones.
- 9.
See Chap. 4 for the definition of conditioning.
- 10.
Prove that \(M_{n} \leq \theta M_{n-1}\).
- 11.
The maximum value of \(\vert \varphi _{n+1}(t) -\varphi _{n}(t)\vert \) is at the midpoint.
- 12.
\(\varphi _{n} =\varphi _{0} +\sum _{ k=0}^{n-1}(\varphi _{k+1} -\varphi _{k})\).
- 13.
Use the MATLAB functions ones, diag.
- 14.
Starting with \(\boldsymbol{x}^{0}\), at each step compute \(\boldsymbol{x}^{1}\) such that \(\boldsymbol{A}\boldsymbol{x}^{1} = h^{2}(\boldsymbol{f} -\sin \boldsymbol{x}^{0}) -\boldsymbol{b}\) where \(\boldsymbol{x}^{0}\) is the previous value and \(\boldsymbol{x}^{1}\) the new one, compute the error \(\|\boldsymbol{x}^{1} -\boldsymbol{x}^{0}\|_{\infty }\) and save it in an array erriter(k) with p + 1 components. Update with \(\boldsymbol{x}^{0} = \boldsymbol{x}^{1}\).
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Lyche, T., Merrien, JL. (2014). Iterative Methods. In: Exercises in Computational Mathematics with MATLAB. Problem Books in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43511-3_5
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