SUBROUTINE DTBT03( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, \$ SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK, \$ RESID ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER DIAG, TRANS, UPLO INTEGER KD, LDAB, LDB, LDX, N, NRHS DOUBLE PRECISION RESID, SCALE, TSCAL * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), CNORM( * ), \$ WORK( * ), X( LDX, * ) * .. * * Purpose * ======= * * DTBT03 computes the residual for the solution to a scaled triangular * system of equations A*x = s*b or A'*x = s*b when A is a * triangular band matrix. Here A' is the transpose of A, s is a scalar, * and x and b are N by NRHS matrices. The test ratio is the maximum * over the number of right hand sides of * norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ), * where op(A) denotes A or A' and EPS is the machine epsilon. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the matrix A is upper or lower triangular. * = 'U': Upper triangular * = 'L': Lower triangular * * TRANS (input) CHARACTER*1 * Specifies the operation applied to A. * = 'N': A *x = b (No transpose) * = 'T': A'*x = b (Transpose) * = 'C': A'*x = b (Conjugate transpose = Transpose) * * DIAG (input) CHARACTER*1 * Specifies whether or not the matrix A is unit triangular. * = 'N': Non-unit triangular * = 'U': Unit triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals or subdiagonals of the * triangular band matrix A. KD >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices X and B. NRHS >= 0. * * AB (input) DOUBLE PRECISION array, dimension (LDAB,N) * The upper or lower triangular band matrix A, stored in the * first kd+1 rows of the array. The j-th column of A is stored * in the j-th column of the array AB as follows: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * SCALE (input) DOUBLE PRECISION * The scaling factor s used in solving the triangular system. * * CNORM (input) DOUBLE PRECISION array, dimension (N) * The 1-norms of the columns of A, not counting the diagonal. * * TSCAL (input) DOUBLE PRECISION * The scaling factor used in computing the 1-norms in CNORM. * CNORM actually contains the column norms of TSCAL*A. * * X (input) DOUBLE PRECISION array, dimension (LDX,NRHS) * The computed solution vectors for the system of linear * equations. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS) * The right hand side vectors for the system of linear * equations. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * WORK (workspace) DOUBLE PRECISION array, dimension (N) * * RESID (output) DOUBLE PRECISION * The maximum over the number of right hand sides of * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE, ZERO PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 ) * .. * .. Local Scalars .. INTEGER IX, J DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX DOUBLE PRECISION DLAMCH EXTERNAL LSAME, IDAMAX, DLAMCH * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DLABAD, DSCAL, DTBMV * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, MAX * .. * .. Executable Statements .. * * Quick exit if N = 0 * IF( N.LE.0 .OR. NRHS.LE.0 ) THEN RESID = ZERO RETURN END IF EPS = DLAMCH( 'Epsilon' ) SMLNUM = DLAMCH( 'Safe minimum' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Compute the norm of the triangular matrix A using the column * norms already computed by DLATBS. * TNORM = ZERO IF( LSAME( DIAG, 'N' ) ) THEN IF( LSAME( UPLO, 'U' ) ) THEN DO 10 J = 1, N TNORM = MAX( TNORM, TSCAL*ABS( AB( KD+1, J ) )+ \$ CNORM( J ) ) 10 CONTINUE ELSE DO 20 J = 1, N TNORM = MAX( TNORM, TSCAL*ABS( AB( 1, J ) )+CNORM( J ) ) 20 CONTINUE END IF ELSE DO 30 J = 1, N TNORM = MAX( TNORM, TSCAL+CNORM( J ) ) 30 CONTINUE END IF * * Compute the maximum over the number of right hand sides of * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ). * RESID = ZERO DO 40 J = 1, NRHS CALL DCOPY( N, X( 1, J ), 1, WORK, 1 ) IX = IDAMAX( N, WORK, 1 ) XNORM = MAX( ONE, ABS( X( IX, J ) ) ) XSCAL = ( ONE / XNORM ) / DBLE( KD+1 ) CALL DSCAL( N, XSCAL, WORK, 1 ) CALL DTBMV( UPLO, TRANS, DIAG, N, KD, AB, LDAB, WORK, 1 ) CALL DAXPY( N, -SCALE*XSCAL, B( 1, J ), 1, WORK, 1 ) IX = IDAMAX( N, WORK, 1 ) ERR = TSCAL*ABS( WORK( IX ) ) IX = IDAMAX( N, X( 1, J ), 1 ) XNORM = ABS( X( IX, J ) ) IF( ERR*SMLNUM.LE.XNORM ) THEN IF( XNORM.GT.ZERO ) \$ ERR = ERR / XNORM ELSE IF( ERR.GT.ZERO ) \$ ERR = ONE / EPS END IF IF( ERR*SMLNUM.LE.TNORM ) THEN IF( TNORM.GT.ZERO ) \$ ERR = ERR / TNORM ELSE IF( ERR.GT.ZERO ) \$ ERR = ONE / EPS END IF RESID = MAX( RESID, ERR ) 40 CONTINUE * RETURN * * End of DTBT03 * END