SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, \$ LDV, TAU, 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 ITYPE, KBAND, LDA, LDU, LDV, N * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ), \$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * ) * .. * * Purpose * ======= * * DSYT21 generally checks a decomposition of the form * * A = U S U' * * where ' means transpose, A is symmetric, U is orthogonal, and S is * diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). * * If ITYPE=1, then U is represented as a dense matrix; otherwise U is * expressed as a product of Householder transformations, whose vectors * are stored in the array "V" and whose scaling constants are in "TAU". * We shall use the letter "V" to refer to the product of Householder * transformations (which should be equal to U). * * Specifically, if ITYPE=1, then: * * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* * RESULT(2) = | I - UU' | / ( n ulp ) * * If ITYPE=2, then: * * RESULT(1) = | A - V S V' | / ( |A| n ulp ) * * If ITYPE=3, then: * * RESULT(1) = | I - VU' | / ( n ulp ) * * For ITYPE > 1, the transformation U is expressed as a product * V = H(1)...H(n-2), where H(j) = I - tau(j) v(j) v(j)' and each * vector v(j) has its first j elements 0 and the remaining n-j elements * stored in V(j+1:n,j). * * Arguments * ========= * * ITYPE (input) INTEGER * Specifies the type of tests to be performed. * 1: U expressed as a dense orthogonal matrix: * RESULT(1) = | A - U S U' | / ( |A| n ulp ) *and* * RESULT(2) = | I - UU' | / ( n ulp ) * * 2: U expressed as a product V of Housholder transformations: * RESULT(1) = | A - V S V' | / ( |A| n ulp ) * * 3: U expressed both as a dense orthogonal matrix and * as a product of Housholder transformations: * RESULT(1) = | I - VU' | / ( n ulp ) * * 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, DSYT21 does nothing. * It must be at least zero. * * KBAND (input) INTEGER * The bandwidth of the matrix. 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 N. * * D (input) DOUBLE PRECISION array, dimension (N) * The diagonal of the (symmetric tri-) diagonal matrix. * * E (input) DOUBLE PRECISION array, dimension (N-1) * The off-diagonal of the (symmetric tri-) diagonal matrix. * E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and * (3,2) element, etc. * Not referenced if KBAND=0. * * U (input) DOUBLE PRECISION array, dimension (LDU, N) * If ITYPE=1 or 3, this contains the orthogonal matrix in * the decomposition, expressed as a dense matrix. If ITYPE=2, * then it is not referenced. * * LDU (input) INTEGER * The leading dimension of U. LDU must be at least N and * at least 1. * * V (input) DOUBLE PRECISION array, dimension (LDV, N) * If ITYPE=2 or 3, the columns of this array contain the * Householder vectors used to describe the orthogonal matrix * in the decomposition. If UPLO='L', then the vectors are in * the lower triangle, if UPLO='U', then in the upper * triangle. * *NOTE* If ITYPE=2 or 3, V is modified and restored. The * subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') * is set to one, and later reset to its original value, during * the course of the calculation. * If ITYPE=1, then it is neither referenced nor modified. * * LDV (input) INTEGER * The leading dimension of V. LDV must be at least N and * at least 1. * * TAU (input) DOUBLE PRECISION array, dimension (N) * If ITYPE >= 2, then TAU(j) is the scalar factor of * v(j) v(j)' in the Householder transformation H(j) of * the product U = H(1)...H(n-2) * If ITYPE < 2, then TAU is not referenced. * * WORK (workspace) DOUBLE PRECISION array, dimension (2*N**2) * * 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. * RESULT(1) is always modified. RESULT(2) is modified only * if ITYPE=1. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE, TEN PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IINFO, J, JCOL, JR, JROW DOUBLE PRECISION ANORM, ULP, UNFL, VSAVE, WNORM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, DLANGE, DLANSY EXTERNAL LSAME, DLAMCH, DLANGE, DLANSY * .. * .. External Subroutines .. EXTERNAL DGEMM, DLACPY, DLARFY, DLASET, DORM2L, DORM2R, \$ DSYR, DSYR2 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * RESULT( 1 ) = ZERO IF( ITYPE.EQ.1 ) \$ RESULT( 2 ) = ZERO IF( N.LE.0 ) \$ RETURN * 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 * IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN RESULT( 1 ) = TEN / ULP RETURN END IF * * Do Test 1 * * Norm of A: * IF( ITYPE.EQ.3 ) THEN ANORM = ONE ELSE ANORM = MAX( DLANSY( '1', CUPLO, N, A, LDA, WORK ), UNFL ) END IF * * Compute error matrix: * IF( ITYPE.EQ.1 ) THEN * * ITYPE=1: error = A - U S U' * CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) CALL DLACPY( CUPLO, N, N, A, LDA, WORK, N ) * DO 10 J = 1, N CALL DSYR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK, N ) 10 CONTINUE * IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN DO 20 J = 1, N - 1 CALL DSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), \$ 1, WORK, N ) 20 CONTINUE END IF WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) ) * ELSE IF( ITYPE.EQ.2 ) THEN * * ITYPE=2: error = V S V' - A * CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) * IF( LOWER ) THEN WORK( N**2 ) = D( N ) DO 40 J = N - 1, 1, -1 IF( KBAND.EQ.1 ) THEN WORK( ( N+1 )*( J-1 )+2 ) = ( ONE-TAU( J ) )*E( J ) DO 30 JR = J + 2, N WORK( ( J-1 )*N+JR ) = -TAU( J )*E( J )*V( JR, J ) 30 CONTINUE END IF * VSAVE = V( J+1, J ) V( J+1, J ) = ONE CALL DLARFY( 'L', N-J, V( J+1, J ), 1, TAU( J ), \$ WORK( ( N+1 )*J+1 ), N, WORK( N**2+1 ) ) V( J+1, J ) = VSAVE WORK( ( N+1 )*( J-1 )+1 ) = D( J ) 40 CONTINUE ELSE WORK( 1 ) = D( 1 ) DO 60 J = 1, N - 1 IF( KBAND.EQ.1 ) THEN WORK( ( N+1 )*J ) = ( ONE-TAU( J ) )*E( J ) DO 50 JR = 1, J - 1 WORK( J*N+JR ) = -TAU( J )*E( J )*V( JR, J+1 ) 50 CONTINUE END IF * VSAVE = V( J, J+1 ) V( J, J+1 ) = ONE CALL DLARFY( 'U', J, V( 1, J+1 ), 1, TAU( J ), WORK, N, \$ WORK( N**2+1 ) ) V( J, J+1 ) = VSAVE WORK( ( N+1 )*J+1 ) = D( J+1 ) 60 CONTINUE END IF * DO 90 JCOL = 1, N IF( LOWER ) THEN DO 70 JROW = JCOL, N WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) ) \$ - A( JROW, JCOL ) 70 CONTINUE ELSE DO 80 JROW = 1, JCOL WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) ) \$ - A( JROW, JCOL ) 80 CONTINUE END IF 90 CONTINUE WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) ) * ELSE IF( ITYPE.EQ.3 ) THEN * * ITYPE=3: error = U V' - I * IF( N.LT.2 ) \$ RETURN CALL DLACPY( ' ', N, N, U, LDU, WORK, N ) IF( LOWER ) THEN CALL DORM2R( 'R', 'T', N, N-1, N-1, V( 2, 1 ), LDV, TAU, \$ WORK( N+1 ), N, WORK( N**2+1 ), IINFO ) ELSE CALL DORM2L( 'R', 'T', N, N-1, N-1, V( 1, 2 ), LDV, TAU, \$ WORK, N, WORK( N**2+1 ), IINFO ) END IF IF( IINFO.NE.0 ) THEN RESULT( 1 ) = TEN / ULP RETURN END IF * DO 100 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 100 CONTINUE * WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) ) END IF * 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 * IF( ITYPE.EQ.1 ) THEN CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK, \$ N ) * DO 110 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 110 CONTINUE * RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, \$ WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP ) END IF * RETURN * * End of DSYT21 * END