Let AER dxd be invertible, let berd, and let X*ER d be the unique solution of the linear system Ax=b. Let BER dxd be an invertible matrix, and consider the iteration x(k+1) = x(k) + B¯¹(b − Ax(k)), KEN. Which additional properties does the matrix B need to possess to ensure that x* is a fixed point of the iteration? a. We must have for some pe[1,00]. The matrix B must be strictly diagonally dominant. O b. O c. The matrix B must be lower triangular. ||I − B¯¹ A||p < 1 O d. O e. We must have ||B||p<1 for some p=[1,00]. O f. None. The vector x* is a fixed point of the iteration for every invertible matrix B. The matrix B must admit a splitting B=D-L-R that conforms with A.

Let AER dxd be invertible, let berd, and let XER d be the unique solution of the linear system Ax=b. Let BER dxd be an invertible matrix, and consider the iteration x(k+1) = x(k) + B¯¹(b − Ax(k)), KEN. Which additional properties does the matrix B need to possess to ensure that x is a fixed point of the iteration? a. We must have for some pe[1,00]. The matrix B must be strictly diagonally dominant. O b. O c. The matrix B must be lower triangular. ||I − B¯¹ A||p < 1 O d. O e. We must have ||B||p<1 for some p=[1,00]. O f. None. The vector x* is a fixed point of the iteration for every invertible matrix B. The matrix B must admit a splitting B=D-L-R that conforms with A.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.2: Direct Methods For Solving Linear Systems

Problem 3BEXP

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