testdata/lapack/TESTING/EIG/ssyt21.f
*> \brief \b SSYT21
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE SSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
* LDV, TAU, WORK, RESULT )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER ITYPE, KBAND, LDA, LDU, LDV, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
* $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SSYT21 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 ) *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 )
*>
*> 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).
*> \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', 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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The size of the matrix. If it is zero, SSYT21 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] A
*> \verbatim
*> A is REAL 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. It must be at least 1
*> and at least N.
*> \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] V
*> \verbatim
*> V is REAL 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.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of V. LDV must be at least N and
*> at least 1.
*> \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 (2*N**2)
*> \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 SSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
$ LDV, 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, LDA, LDU, LDV, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TEN
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0, TEN = 10.0E0 )
* ..
* .. Local Scalars ..
LOGICAL LOWER
CHARACTER CUPLO
INTEGER IINFO, J, JCOL, JR, JROW
REAL ANORM, ULP, UNFL, VSAVE, WNORM
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH, SLANGE, SLANSY
EXTERNAL LSAME, SLAMCH, SLANGE, SLANSY
* ..
* .. External Subroutines ..
EXTERNAL SGEMM, SLACPY, SLARFY, SLASET, SORM2L, SORM2R,
$ SSYR, SSYR2
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. 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 = 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( SLANSY( '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 SLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
CALL SLACPY( CUPLO, N, N, A, LDA, WORK, N )
*
DO 10 J = 1, N
CALL SSYR( 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 SSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
$ 1, WORK, N )
20 CONTINUE
END IF
WNORM = SLANSY( '1', CUPLO, N, WORK, N, 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( 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 SLARFY( '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 SLARFY( '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 = SLANSY( '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 SLACPY( ' ', N, N, U, LDU, WORK, N )
IF( LOWER ) THEN
CALL SORM2R( 'R', 'T', N, N-1, N-1, V( 2, 1 ), LDV, TAU,
$ WORK( N+1 ), N, WORK( N**2+1 ), IINFO )
ELSE
CALL SORM2L( '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 = 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 110 J = 1, N
WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
110 CONTINUE
*
RESULT( 2 ) = MIN( SLANGE( '1', N, N, WORK, N,
$ WORK( N**2+1 ) ), REAL( N ) ) / ( N*ULP )
END IF
*
RETURN
*
* End of SSYT21
*
END