Let Ax = b be the system to be solved, and the computed solution. Let n be the dimension of A. An approximate error bound for may be obtained in one of the following two ways, depending on whether the solution is computed by a simple driver or an expert driver:
can be computed by the following code fragment.
EPSMCH = SLAMCH( 'E' ) * Get infinity-norm of A ANORM = SLANGE( 'I', N, N, A, LDA, WORK ) * Solve system; The solution X overwrites B CALL SGESV( N, 1, A, LDA, IPIV, B, LDB, INFO ) IF( INFO.GT.0 ) THEN PRINT *,'Singular Matrix' ELSE IF (N .GT. 0) THEN * Get reciprocal condition number RCOND of A CALL SGECON( 'I', N, A, LDA, ANORM, RCOND, $ WORK, IWORK, INFO ) RCOND = MAX( RCOND, EPSMCH ) ERRBD = EPSMCH / RCOND END IF
For example, suppose
Then (to 4 decimal places)
, , the true reciprocal condition number , , and the true error .
For example, the following code fragment solves Ax = b and computes an approximate error bound FERR:
CALL SGESVX( 'E', 'N', N, 1, A, LDA, AF, LDAF, IPIV, $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, $ WORK, IWORK, INFO ) IF( INFO.GT.0 ) PRINT *,'(Nearly) Singular Matrix'
For the same A and b as above, , , and the actual error is .
This example illustrates that the expert driver provides an error bound with less programming effort than the simple driver, and also that it may produce a significantly more accurate answer.
Similar code fragments, with obvious adaptations, may be used with all the driver routines for linear equations listed in Table 2.2. For example, if a symmetric system is solved using the simple driver xSYSV, then xLANSY must be used to compute ANORM, and xSYCON must be used to compute RCOND.