SUBROUTINE DSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, \$ 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, LDU, N * .. * .. Array Arguments .. DOUBLE PRECISION AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ), \$ U( LDU, * ), VP( * ), WORK( * ) * .. * * Purpose * ======= * * DSPT21 generally checks a decomposition of the form * * A = U S U' * * where ' means transpose, A is symmetric (stored in packed format), 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 the 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 ) * * Packed storage means that, for example, if UPLO='U', then the columns * of the upper triangle of A are stored one after another, so that * A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if * UPLO='L', then the columns of the lower triangle of A are stored one * after another in AP, so that A(j+1,j+1) immediately follows A(n,j) * in the array AP. This means that A(i,j) is stored in: * * AP( i + j*(j-1)/2 ) if UPLO='U' * * AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L' * * The array VP bears the same relation to the matrix V that A does to * AP. * * For ITYPE > 1, the transformation U is expressed as a product * of Householder transformations: * * If UPLO='U', then V = H(n-1)...H(1), where * * H(j) = I - tau(j) v(j) v(j)' * * and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), * (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ), * the j-th element is 1, and the last n-j elements are 0. * * If UPLO='L', then V = H(1)...H(n-1), where * * H(j) = I - tau(j) v(j) v(j)' * * and the first j elements of v(j) are 0, the (j+1)-st is 1, and the * (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., * in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .) * * 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', AP and VP are considered to contain the upper * triangle of A and V. * If UPLO='L', AP and VP are considered to contain the lower * triangle of A and V. * * N (input) INTEGER * The size of the matrix. If it is zero, DSPT21 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. * * AP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) * The original (unfactored) matrix. It is assumed to be * symmetric, and contains the columns of just the upper * triangle (UPLO='U') or only the lower triangle (UPLO='L'), * packed one after another. * * 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. * * VP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2) * If ITYPE=2 or 3, the columns of this array contain the * Householder vectors used to describe the orthogonal matrix * in the decomposition, as described in purpose. * *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. * * 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 (N**2+N) * Workspace. * * 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 ) DOUBLE PRECISION HALF PARAMETER ( HALF = 1.0D+0 / 2.0D+0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IINFO, J, JP, JP1, JR, LAP DOUBLE PRECISION ANORM, TEMP, ULP, UNFL, VSAVE, WNORM * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DDOT, DLAMCH, DLANGE, DLANSP EXTERNAL LSAME, DDOT, DLAMCH, DLANGE, DLANSP * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMM, DLACPY, DLASET, DOPMTR, \$ DSPMV, DSPR, DSPR2 * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * * 1) Constants * RESULT( 1 ) = ZERO IF( ITYPE.EQ.1 ) \$ RESULT( 2 ) = ZERO IF( N.LE.0 ) \$ RETURN * LAP = ( 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 * 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( DLANSP( '1', CUPLO, N, AP, 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 DCOPY( LAP, AP, 1, WORK, 1 ) * DO 10 J = 1, N CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK ) 10 CONTINUE * IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN DO 20 J = 1, N - 1 CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), \$ 1, WORK ) 20 CONTINUE END IF WNORM = DLANSP( '1', CUPLO, N, WORK, 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( LAP ) = D( N ) DO 40 J = N - 1, 1, -1 JP = ( ( 2*N-J )*( J-1 ) ) / 2 JP1 = JP + N - J IF( KBAND.EQ.1 ) THEN WORK( JP+J+1 ) = ( ONE-TAU( J ) )*E( J ) DO 30 JR = J + 2, N WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR ) 30 CONTINUE END IF * IF( TAU( J ).NE.ZERO ) THEN VSAVE = VP( JP+J+1 ) VP( JP+J+1 ) = ONE CALL DSPMV( 'L', N-J, ONE, WORK( JP1+J+1 ), \$ VP( JP+J+1 ), 1, ZERO, WORK( LAP+1 ), 1 ) TEMP = -HALF*TAU( J )*DDOT( N-J, WORK( LAP+1 ), 1, \$ VP( JP+J+1 ), 1 ) CALL DAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ), \$ 1 ) CALL DSPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1, \$ WORK( LAP+1 ), 1, WORK( JP1+J+1 ) ) VP( JP+J+1 ) = VSAVE END IF WORK( JP+J ) = D( J ) 40 CONTINUE ELSE WORK( 1 ) = D( 1 ) DO 60 J = 1, N - 1 JP = ( J*( J-1 ) ) / 2 JP1 = JP + J IF( KBAND.EQ.1 ) THEN WORK( JP1+J ) = ( ONE-TAU( J ) )*E( J ) DO 50 JR = 1, J - 1 WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR ) 50 CONTINUE END IF * IF( TAU( J ).NE.ZERO ) THEN VSAVE = VP( JP1+J ) VP( JP1+J ) = ONE CALL DSPMV( 'U', J, ONE, WORK, VP( JP1+1 ), 1, ZERO, \$ WORK( LAP+1 ), 1 ) TEMP = -HALF*TAU( J )*DDOT( J, WORK( LAP+1 ), 1, \$ VP( JP1+1 ), 1 ) CALL DAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ), \$ 1 ) CALL DSPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1, \$ WORK( LAP+1 ), 1, WORK ) VP( JP1+J ) = VSAVE END IF WORK( JP1+J+1 ) = D( J+1 ) 60 CONTINUE END IF * DO 70 J = 1, LAP WORK( J ) = WORK( J ) - AP( J ) 70 CONTINUE WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LAP+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 ) CALL DOPMTR( 'R', CUPLO, 'T', N, N, VP, TAU, WORK, N, \$ WORK( N**2+1 ), IINFO ) IF( IINFO.NE.0 ) THEN RESULT( 1 ) = TEN / ULP RETURN END IF * DO 80 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 80 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 90 J = 1, N WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE 90 CONTINUE * RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N, \$ WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP ) END IF * RETURN * * End of DSPT21 * END