Residual-Based Simpler Block GMRES for Nonsymmetric Linear Systems with Multiple Right-Hand Sides

Algorithm 4

RB-SBGMRES-D(m).

Given , set . If , accept and exit.

Compute the QR factorization of .

Determine deflation unitary matrix and such that (see Algorithm 5), and set .

Define , with as the first columns of , and define .

For

  Compute .

  If

  Compute the QR factorization of .

  Determine deflation unitary matrix and such that (see Algorithm 5), and set .

  Define , with as the first columns of , and define .

  Else

  For

 .

 .

 End For

   .

   .

  Define , compute the QR factorization of , leading to

 , with defined as (41).

  Determine deflation unitary matrix by the SVD of ,

 and such that (see Algorithm 5), and set .

  Define , with as the first

 columns of , and set , .

 End If

  Compute , .

  If , break.

  Compute the QR factorization of .

  Determine deflation unitary matrix and such that (see Algorithm 5), and set .

  Define , with as the first

 columns of , and define .

 End For

Solve the triangular system for , with

From the approximate solution

If , then accept and exit; otherwise, restart: set and go to Step (1).