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 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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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].
O b. The matrix B must be strictly diagonally dominant.
c. The matrix B must be lower triangular.
||I – B¯¹ A||₂ < 1
d.
The matrix B must admit a splitting B-D-L-R that conforms with A.
O e. We must have ||B||p<1 for some pe[1,00].
O f. None. The vector x* is a fixed point of the iteration for every invertible matrix B.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe1fc35-672a-49fd-831b-9642c77888ed%2F0054ae45-49e3-44b4-8704-9245b1f01219%2F7phuf9_processed.png&w=3840&q=75)
Transcribed Image Text: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].
O b. The matrix B must be strictly diagonally dominant.
c. The matrix B must be lower triangular.
||I – B¯¹ A||₂ < 1
d.
The matrix B must admit a splitting B-D-L-R that conforms with A.
O e. We must have ||B||p<1 for some pe[1,00].
O f. None. The vector x* is a fixed point of the iteration for every invertible matrix B.
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