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The goal of this problem is to prove that, when the conjugate-gradient method is used to solve Ax = b, the norms of the errors decrease monotonically. %3D (i) Prove that the conjugate directions satisfy pPj >0 for all i and j. (ii) Prove that the estimates of the solution satisfy ||x; || > ||xi-1|| for all i. (Here the 2-norm is used.) (iii) Let x, solve Ax b. Prove that ||x; – x || < ||X;-1 – x || for all i.

The goal of this problem is to prove that, when the conjugate-gradient method is used to solve Ax = b, the norms of the errors decrease monotonically. %3D (i) Prove that the conjugate directions satisfy pPj >0 for all i and j. (ii) Prove that the estimates of the solution satisfy ||x; || > ||xi-1|| for all i. (Here the 2-norm is used.) (iii) Let x, solve Ax b. Prove that ||x; – x || < ||X;-1 – x || for all i.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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**Problem 2.12: Conjugate Gradient Method** The aim of this problem is to prove that, when the conjugate-gradient method is used to solve the equation \( Ax = b \), the norms of the errors decrease monotonically. The problem is divided into the following parts: (i) **Conjugate Directions:** Prove that the conjugate directions satisfy \( p_i^T p_j \geq 0 \) for all \( i \) and \( j \). (ii) **Estimates of the Solution:** Prove that the estimates of the solution satisfy \( \|x_i\| \geq \|x_{i-1}\| \) for all \( i \). (Here, the 2-norm is used.) (iii) **Solution Error Norms:** Let \( x_* \) solve \( Ax = b \). Prove that \( \|x_i - x_*\| \leq \|x_{i-1} - x_*\| \) for all \( i \).
Transcribed Image Text:**Problem 2.12: Conjugate Gradient Method** The aim of this problem is to prove that, when the conjugate-gradient method is used to solve the equation \( Ax = b \), the norms of the errors decrease monotonically. The problem is divided into the following parts: (i) **Conjugate Directions:** Prove that the conjugate directions satisfy \( p_i^T p_j \geq 0 \) for all \( i \) and \( j \). (ii) **Estimates of the Solution:** Prove that the estimates of the solution satisfy \( \|x_i\| \geq \|x_{i-1}\| \) for all \( i \). (Here, the 2-norm is used.) (iii) **Solution Error Norms:** Let \( x_* \) solve \( Ax = b \). Prove that \( \|x_i - x_*\| \leq \|x_{i-1} - x_*\| \) for all \( i \).
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