Let A be a general non symmetric nonsingular square matrix, and consider the following two…

Let A be a
general non symmetric nonsingular square matrix, and consider the following two
alternatives. The first is applying GMRES to solve the linear system Ax = b;
the second is applying CG to the normal equations

We briefly
discussed this in Section 7.5; the method we mentioned in that context was
CGLS.

(a) Suppose
your matrix A is nearly orthogonal. Which of the two solvers is expected to
converge faster?

(b) Suppose
your matrix is block diagonal relative to 2 ×2 blocks, where the jth block is
given by

with j =
1,…,n/2. Which of the two solvers is expected to converge
»

Let A be a
general non symmetric nonsingular square matrix, and consider the following two
alternatives. The first is applying GMRES to solve the linear system Ax = b;
the second is applying CG to the normal equations

We briefly
discussed this in Section 7.5; the method we mentioned in that context was
CGLS.

(a) Suppose
your matrix A is nearly orthogonal. Which of the two solvers is expected to
converge faster?

(b) Suppose
your matrix is block diagonal relative to 2 ×2 blocks, where the jth block is
given by

with j =
1,…,n/2. Which of the two solvers is expected to converge faster?

[Hint:
Consider the eigenvalues and the singular values of the matrices.]

»

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