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SUBROUTINE DSBT21( UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK,
$ RESULT )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* February 29, 1992
*
* .. 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
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