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

Algorithm 3

SBGMRES(m).

Given , set . If , accept and exit.

For

  Compute if , for .

  For

 .

 .

 End For

  Compute the QR factorization of .

  Compute , .

  If , break.

 End For

Solve the triangular system for .

Form the approximate solution

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