SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, \$ RESULT ) * * -- LAPACK test routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER KA, KS, LDA, LDU, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), \$ U( LDU, * ), WORK( * ) * .. * * Purpose * ======= * * DSBT21 generally checks a decomposition of the form * * A = U S U' * * where ' means transpose, A is symmetric banded, U is * orthogonal, and S is diagonal (if KS=0) or symmetric * tridiagonal (if KS=1). * * Specifically: * * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* * RESULT(2) = | I - UU' | / ( n ulp ) * * Arguments * ========= * * UPLO (input) CHARACTER * If UPLO='U', the upper triangle of A and V will be used and * the (strictly) lower triangle will not be referenced. * If UPLO='L', the lower triangle of A and V will be used and * the (strictly) upper triangle will not be referenced. * * N (input) INTEGER * The size of the matrix. If it is zero, DSBT21 does nothing. * It must be at least zero. * * KA (input) INTEGER * The bandwidth of the matrix A. It must be at least zero. If * it is larger than N-1, then max( 0, N-1 ) will be used. * * KS (input) INTEGER * The bandwidth of the matrix S. It may only be zero or one. * If zero, then S is diagonal, and E is not referenced. If * one, then S is symmetric tri-diagonal. * * A (input) DOUBLE PRECISION array, dimension (LDA, N) * The original (unfactored) matrix. It is assumed to be * symmetric, and only the upper (UPLO='U') or only the lower * (UPLO='L') will be referenced. * * LDA (input) INTEGER * The leading dimension of A. It must be at least 1 * and at least min( KA, N-1 ). * * D (input) DOUBLE PRECISION array, dimension (N) * The diagonal of the (symmetric tri-) diagonal matrix S. * * E (input) DOUBLE PRECISION array, dimension (N-1) * The off-diagonal of the (symmetric tri-) diagonal matrix S. * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and * (3,2) element, etc. * Not referenced if KS=0. * * U (input) DOUBLE PRECISION array, dimension (LDU, N) * The orthogonal matrix in the decomposition, expressed as a * dense matrix (i.e., not as a product of Householder * transformations, Givens transformations, etc.) * * LDU (input) INTEGER * The leading dimension of U. LDU must be at least N and * at least 1. * * WORK (workspace) DOUBLE PRECISION array, dimension (N**2+N) * * RESULT (output) DOUBLE PRECISION array, dimension (2) * The values computed by the two tests described above. The * values are currently limited to 1/ulp, to avoid overflow. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IKA, J, JC, JR, LW DOUBLE PRECISION ANORM, ULP, UNFL, WNORM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE, DLANSB, DLANSP EXTERNAL LSAME, DLAMCH, DLANGE, DLANSB, DLANSP * .. * .. External Subroutines .. EXTERNAL DGEMM, DSPR, DSPR2 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * * Constants * RESULT( 1 ) = ZERO RESULT( 2 ) = ZERO IF( N.LE.0 ) \$ RETURN * IKA = MAX( 0, MIN( N-1, KA ) ) LW = ( N*( N+1 ) ) / 2 * IF( LSAME( UPLO, 'U' ) ) THEN LOWER = .FALSE. CUPLO = 'U' ELSE LOWER = .TRUE. CUPLO = 'L' END IF * UNFL = DLAMCH( 'Safe minimum' ) ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' ) * * Some Error Checks * * Do Test 1 * * Norm of A: * ANORM = MAX( DLANSB( '1', CUPLO, N, IKA, A, LDA, WORK ), UNFL ) * * Compute error matrix: Error = A - U S U' * * Copy A from SB to SP storage format. * J = 0 DO 50 JC = 1, N IF( LOWER ) THEN DO 10 JR = 1, MIN( IKA+1, N+1-JC ) J = J + 1 WORK( J ) = A( JR, JC ) 10 CONTINUE DO 20 JR = IKA + 2, N + 1 - JC J = J + 1 WORK( J ) = ZERO 20 CONTINUE ELSE DO 30 JR = IKA + 2, JC J = J + 1 WORK( J ) = ZERO 30 CONTINUE DO 40 JR = MIN( IKA, JC-1 ), 0, -1 J = J + 1 WORK( J ) = A( IKA+1-JR, JC ) 40 CONTINUE END IF 50 CONTINUE * DO 60 J = 1, N CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK ) 60 CONTINUE * IF( N.GT.1 .AND. KS.EQ.1 ) THEN DO 70 J = 1, N - 1 CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), 1, \$ WORK ) 70 CONTINUE END IF WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LW+1 ) ) * IF( ANORM.GT.WNORM ) THEN RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP ) ELSE IF( ANORM.LT.ONE ) THEN RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP ) ELSE RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP ) END IF END IF * * Do Test 2 * * Compute UU' - I * CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK, \$ N ) * DO 80 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 80 CONTINUE * RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ), \$ DBLE( N ) ) / ( N*ULP ) * RETURN * * End of DSBT21 * END