*> \brief \b SSPT21 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE SSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, * TAU, WORK, RESULT ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER ITYPE, KBAND, LDU, N * .. * .. Array Arguments .. * REAL AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ), * $ U( LDU, * ), VP( * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SSPT21 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 ) *andC> 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 ) .) *> \endverbatim * * Arguments: * ========== * *> \param[in] ITYPE *> \verbatim *> ITYPE is 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 ) *andC> 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 ) *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is 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. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The size of the matrix. If it is zero, SSPT21 does nothing. *> It must be at least zero. *> \endverbatim *> *> \param[in] KBAND *> \verbatim *> KBAND is 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. *> \endverbatim *> *> \param[in] AP *> \verbatim *> AP is REAL 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. *> \endverbatim *> *> \param[in] D *> \verbatim *> D is REAL array, dimension (N) *> The diagonal of the (symmetric tri-) diagonal matrix. *> \endverbatim *> *> \param[in] E *> \verbatim *> E is REAL 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. *> \endverbatim *> *> \param[in] U *> \verbatim *> U is REAL 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. *> \endverbatim *> *> \param[in] LDU *> \verbatim *> LDU is INTEGER *> The leading dimension of U. LDU must be at least N and *> at least 1. *> \endverbatim *> *> \param[in] VP *> \verbatim *> VP is REAL 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. *> \endverbatim *> *> \param[in] TAU *> \verbatim *> TAU is REAL 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. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is REAL array, dimension (N**2+N) *> Workspace. *> \endverbatim *> *> \param[out] RESULT *> \verbatim *> RESULT is REAL 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. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup single_eig * * ===================================================================== SUBROUTINE SSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, $ TAU, WORK, RESULT ) * * -- LAPACK test routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER UPLO INTEGER ITYPE, KBAND, LDU, N * .. * .. Array Arguments .. REAL AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ), $ U( LDU, * ), VP( * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE, TEN PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TEN = 10.0E0 ) REAL HALF PARAMETER ( HALF = 1.0E+0 / 2.0E+0 ) * .. * .. Local Scalars .. LOGICAL LOWER CHARACTER CUPLO INTEGER IINFO, J, JP, JP1, JR, LAP REAL ANORM, TEMP, ULP, UNFL, VSAVE, WNORM * .. * .. External Functions .. LOGICAL LSAME REAL SDOT, SLAMCH, SLANGE, SLANSP EXTERNAL LSAME, SDOT, SLAMCH, SLANGE, SLANSP * .. * .. External Subroutines .. EXTERNAL SAXPY, SCOPY, SGEMM, SLACPY, SLASET, SOPMTR, $ SSPMV, SSPR, SSPR2 * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN, REAL * .. * .. 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 = SLAMCH( 'Safe minimum' ) ULP = SLAMCH( 'Epsilon' )*SLAMCH( '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( SLANSP( '1', CUPLO, N, AP, WORK ), UNFL ) END IF * * Compute error matrix: * IF( ITYPE.EQ.1 ) THEN * * ITYPE=1: error = A - U S U' * CALL SLASET( 'Full', N, N, ZERO, ZERO, WORK, N ) CALL SCOPY( LAP, AP, 1, WORK, 1 ) * DO 10 J = 1, N CALL SSPR( 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 SSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ), $ 1, WORK ) 20 CONTINUE END IF WNORM = SLANSP( '1', CUPLO, N, WORK, WORK( N**2+1 ) ) * ELSE IF( ITYPE.EQ.2 ) THEN * * ITYPE=2: error = V S V' - A * CALL SLASET( '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 SSPMV( 'L', N-J, ONE, WORK( JP1+J+1 ), $ VP( JP+J+1 ), 1, ZERO, WORK( LAP+1 ), 1 ) TEMP = -HALF*TAU( J )*SDOT( N-J, WORK( LAP+1 ), 1, $ VP( JP+J+1 ), 1 ) CALL SAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ), $ 1 ) CALL SSPR2( '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 SSPMV( 'U', J, ONE, WORK, VP( JP1+1 ), 1, ZERO, $ WORK( LAP+1 ), 1 ) TEMP = -HALF*TAU( J )*SDOT( J, WORK( LAP+1 ), 1, $ VP( JP1+1 ), 1 ) CALL SAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ), $ 1 ) CALL SSPR2( '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 = SLANSP( '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 SLACPY( ' ', N, N, U, LDU, WORK, N ) CALL SOPMTR( '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 = SLANGE( '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, REAL( N ) ) / ( N*ULP ) END IF END IF * * Do Test 2 * * Compute UU' - I * IF( ITYPE.EQ.1 ) THEN CALL SGEMM( '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( SLANGE( '1', N, N, WORK, N, $ WORK( N**2+1 ) ), REAL( N ) ) / ( N*ULP ) END IF * RETURN * * End of SSPT21 * END