LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsyt21.f
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00001 *> \brief \b DSYT21
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
00012 *                          LDV, TAU, WORK, RESULT )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            ITYPE, KBAND, LDA, LDU, LDV, N
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
00020 *      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> DSYT21 generally checks a decomposition of the form
00030 *>
00031 *>    A = U S U'
00032 *>
00033 *> where ' means transpose, A is symmetric, U is orthogonal, and S is
00034 *> diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
00035 *>
00036 *> If ITYPE=1, then U is represented as a dense matrix; otherwise U is
00037 *> expressed as a product of Householder transformations, whose vectors
00038 *> are stored in the array "V" and whose scaling constants are in "TAU".
00039 *> We shall use the letter "V" to refer to the product of Householder
00040 *> transformations (which should be equal to U).
00041 *>
00042 *> Specifically, if ITYPE=1, then:
00043 *>
00044 *>    RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC>    RESULT(2) = | I - UU' | / ( n ulp )
00045 *>
00046 *> If ITYPE=2, then:
00047 *>
00048 *>    RESULT(1) = | A - V S V' | / ( |A| n ulp )
00049 *>
00050 *> If ITYPE=3, then:
00051 *>
00052 *>    RESULT(1) = | I - VU' | / ( n ulp )
00053 *>
00054 *> For ITYPE > 1, the transformation U is expressed as a product
00055 *> V = H(1)...H(n-2),  where H(j) = I  -  tau(j) v(j) v(j)' and each
00056 *> vector v(j) has its first j elements 0 and the remaining n-j elements
00057 *> stored in V(j+1:n,j).
00058 *> \endverbatim
00059 *
00060 *  Arguments:
00061 *  ==========
00062 *
00063 *> \param[in] ITYPE
00064 *> \verbatim
00065 *>          ITYPE is INTEGER
00066 *>          Specifies the type of tests to be performed.
00067 *>          1: U expressed as a dense orthogonal matrix:
00068 *>             RESULT(1) = | A - U S U' | / ( |A| n ulp )   *andC>             RESULT(2) = | I - UU' | / ( n ulp )
00069 *>
00070 *>          2: U expressed as a product V of Housholder transformations:
00071 *>             RESULT(1) = | A - V S V' | / ( |A| n ulp )
00072 *>
00073 *>          3: U expressed both as a dense orthogonal matrix and
00074 *>             as a product of Housholder transformations:
00075 *>             RESULT(1) = | I - VU' | / ( n ulp )
00076 *> \endverbatim
00077 *>
00078 *> \param[in] UPLO
00079 *> \verbatim
00080 *>          UPLO is CHARACTER
00081 *>          If UPLO='U', the upper triangle of A and V will be used and
00082 *>          the (strictly) lower triangle will not be referenced.
00083 *>          If UPLO='L', the lower triangle of A and V will be used and
00084 *>          the (strictly) upper triangle will not be referenced.
00085 *> \endverbatim
00086 *>
00087 *> \param[in] N
00088 *> \verbatim
00089 *>          N is INTEGER
00090 *>          The size of the matrix.  If it is zero, DSYT21 does nothing.
00091 *>          It must be at least zero.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] KBAND
00095 *> \verbatim
00096 *>          KBAND is INTEGER
00097 *>          The bandwidth of the matrix.  It may only be zero or one.
00098 *>          If zero, then S is diagonal, and E is not referenced.  If
00099 *>          one, then S is symmetric tri-diagonal.
00100 *> \endverbatim
00101 *>
00102 *> \param[in] A
00103 *> \verbatim
00104 *>          A is DOUBLE PRECISION array, dimension (LDA, N)
00105 *>          The original (unfactored) matrix.  It is assumed to be
00106 *>          symmetric, and only the upper (UPLO='U') or only the lower
00107 *>          (UPLO='L') will be referenced.
00108 *> \endverbatim
00109 *>
00110 *> \param[in] LDA
00111 *> \verbatim
00112 *>          LDA is INTEGER
00113 *>          The leading dimension of A.  It must be at least 1
00114 *>          and at least N.
00115 *> \endverbatim
00116 *>
00117 *> \param[in] D
00118 *> \verbatim
00119 *>          D is DOUBLE PRECISION array, dimension (N)
00120 *>          The diagonal of the (symmetric tri-) diagonal matrix.
00121 *> \endverbatim
00122 *>
00123 *> \param[in] E
00124 *> \verbatim
00125 *>          E is DOUBLE PRECISION array, dimension (N-1)
00126 *>          The off-diagonal of the (symmetric tri-) diagonal matrix.
00127 *>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
00128 *>          (3,2) element, etc.
00129 *>          Not referenced if KBAND=0.
00130 *> \endverbatim
00131 *>
00132 *> \param[in] U
00133 *> \verbatim
00134 *>          U is DOUBLE PRECISION array, dimension (LDU, N)
00135 *>          If ITYPE=1 or 3, this contains the orthogonal matrix in
00136 *>          the decomposition, expressed as a dense matrix.  If ITYPE=2,
00137 *>          then it is not referenced.
00138 *> \endverbatim
00139 *>
00140 *> \param[in] LDU
00141 *> \verbatim
00142 *>          LDU is INTEGER
00143 *>          The leading dimension of U.  LDU must be at least N and
00144 *>          at least 1.
00145 *> \endverbatim
00146 *>
00147 *> \param[in] V
00148 *> \verbatim
00149 *>          V is DOUBLE PRECISION array, dimension (LDV, N)
00150 *>          If ITYPE=2 or 3, the columns of this array contain the
00151 *>          Householder vectors used to describe the orthogonal matrix
00152 *>          in the decomposition.  If UPLO='L', then the vectors are in
00153 *>          the lower triangle, if UPLO='U', then in the upper
00154 *>          triangle.
00155 *>          *NOTE* If ITYPE=2 or 3, V is modified and restored.  The
00156 *>          subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
00157 *>          is set to one, and later reset to its original value, during
00158 *>          the course of the calculation.
00159 *>          If ITYPE=1, then it is neither referenced nor modified.
00160 *> \endverbatim
00161 *>
00162 *> \param[in] LDV
00163 *> \verbatim
00164 *>          LDV is INTEGER
00165 *>          The leading dimension of V.  LDV must be at least N and
00166 *>          at least 1.
00167 *> \endverbatim
00168 *>
00169 *> \param[in] TAU
00170 *> \verbatim
00171 *>          TAU is DOUBLE PRECISION array, dimension (N)
00172 *>          If ITYPE >= 2, then TAU(j) is the scalar factor of
00173 *>          v(j) v(j)' in the Householder transformation H(j) of
00174 *>          the product  U = H(1)...H(n-2)
00175 *>          If ITYPE < 2, then TAU is not referenced.
00176 *> \endverbatim
00177 *>
00178 *> \param[out] WORK
00179 *> \verbatim
00180 *>          WORK is DOUBLE PRECISION array, dimension (2*N**2)
00181 *> \endverbatim
00182 *>
00183 *> \param[out] RESULT
00184 *> \verbatim
00185 *>          RESULT is DOUBLE PRECISION array, dimension (2)
00186 *>          The values computed by the two tests described above.  The
00187 *>          values are currently limited to 1/ulp, to avoid overflow.
00188 *>          RESULT(1) is always modified.  RESULT(2) is modified only
00189 *>          if ITYPE=1.
00190 *> \endverbatim
00191 *
00192 *  Authors:
00193 *  ========
00194 *
00195 *> \author Univ. of Tennessee 
00196 *> \author Univ. of California Berkeley 
00197 *> \author Univ. of Colorado Denver 
00198 *> \author NAG Ltd. 
00199 *
00200 *> \date November 2011
00201 *
00202 *> \ingroup double_eig
00203 *
00204 *  =====================================================================
00205       SUBROUTINE DSYT21( ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V,
00206      $                   LDV, TAU, WORK, RESULT )
00207 *
00208 *  -- LAPACK test routine (version 3.4.0) --
00209 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00210 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00211 *     November 2011
00212 *
00213 *     .. Scalar Arguments ..
00214       CHARACTER          UPLO
00215       INTEGER            ITYPE, KBAND, LDA, LDU, LDV, N
00216 *     ..
00217 *     .. Array Arguments ..
00218       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
00219      $                   TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
00220 *     ..
00221 *
00222 *  =====================================================================
00223 *
00224 *     .. Parameters ..
00225       DOUBLE PRECISION   ZERO, ONE, TEN
00226       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 )
00227 *     ..
00228 *     .. Local Scalars ..
00229       LOGICAL            LOWER
00230       CHARACTER          CUPLO
00231       INTEGER            IINFO, J, JCOL, JR, JROW
00232       DOUBLE PRECISION   ANORM, ULP, UNFL, VSAVE, WNORM
00233 *     ..
00234 *     .. External Functions ..
00235       LOGICAL            LSAME
00236       DOUBLE PRECISION   DLAMCH, DLANGE, DLANSY
00237       EXTERNAL           LSAME, DLAMCH, DLANGE, DLANSY
00238 *     ..
00239 *     .. External Subroutines ..
00240       EXTERNAL           DGEMM, DLACPY, DLARFY, DLASET, DORM2L, DORM2R,
00241      $                   DSYR, DSYR2
00242 *     ..
00243 *     .. Intrinsic Functions ..
00244       INTRINSIC          DBLE, MAX, MIN
00245 *     ..
00246 *     .. Executable Statements ..
00247 *
00248       RESULT( 1 ) = ZERO
00249       IF( ITYPE.EQ.1 )
00250      $   RESULT( 2 ) = ZERO
00251       IF( N.LE.0 )
00252      $   RETURN
00253 *
00254       IF( LSAME( UPLO, 'U' ) ) THEN
00255          LOWER = .FALSE.
00256          CUPLO = 'U'
00257       ELSE
00258          LOWER = .TRUE.
00259          CUPLO = 'L'
00260       END IF
00261 *
00262       UNFL = DLAMCH( 'Safe minimum' )
00263       ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
00264 *
00265 *     Some Error Checks
00266 *
00267       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
00268          RESULT( 1 ) = TEN / ULP
00269          RETURN
00270       END IF
00271 *
00272 *     Do Test 1
00273 *
00274 *     Norm of A:
00275 *
00276       IF( ITYPE.EQ.3 ) THEN
00277          ANORM = ONE
00278       ELSE
00279          ANORM = MAX( DLANSY( '1', CUPLO, N, A, LDA, WORK ), UNFL )
00280       END IF
00281 *
00282 *     Compute error matrix:
00283 *
00284       IF( ITYPE.EQ.1 ) THEN
00285 *
00286 *        ITYPE=1: error = A - U S U'
00287 *
00288          CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
00289          CALL DLACPY( CUPLO, N, N, A, LDA, WORK, N )
00290 *
00291          DO 10 J = 1, N
00292             CALL DSYR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK, N )
00293    10    CONTINUE
00294 *
00295          IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
00296             DO 20 J = 1, N - 1
00297                CALL DSYR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
00298      $                     1, WORK, N )
00299    20       CONTINUE
00300          END IF
00301          WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) )
00302 *
00303       ELSE IF( ITYPE.EQ.2 ) THEN
00304 *
00305 *        ITYPE=2: error = V S V' - A
00306 *
00307          CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
00308 *
00309          IF( LOWER ) THEN
00310             WORK( N**2 ) = D( N )
00311             DO 40 J = N - 1, 1, -1
00312                IF( KBAND.EQ.1 ) THEN
00313                   WORK( ( N+1 )*( J-1 )+2 ) = ( ONE-TAU( J ) )*E( J )
00314                   DO 30 JR = J + 2, N
00315                      WORK( ( J-1 )*N+JR ) = -TAU( J )*E( J )*V( JR, J )
00316    30             CONTINUE
00317                END IF
00318 *
00319                VSAVE = V( J+1, J )
00320                V( J+1, J ) = ONE
00321                CALL DLARFY( 'L', N-J, V( J+1, J ), 1, TAU( J ),
00322      $                      WORK( ( N+1 )*J+1 ), N, WORK( N**2+1 ) )
00323                V( J+1, J ) = VSAVE
00324                WORK( ( N+1 )*( J-1 )+1 ) = D( J )
00325    40       CONTINUE
00326          ELSE
00327             WORK( 1 ) = D( 1 )
00328             DO 60 J = 1, N - 1
00329                IF( KBAND.EQ.1 ) THEN
00330                   WORK( ( N+1 )*J ) = ( ONE-TAU( J ) )*E( J )
00331                   DO 50 JR = 1, J - 1
00332                      WORK( J*N+JR ) = -TAU( J )*E( J )*V( JR, J+1 )
00333    50             CONTINUE
00334                END IF
00335 *
00336                VSAVE = V( J, J+1 )
00337                V( J, J+1 ) = ONE
00338                CALL DLARFY( 'U', J, V( 1, J+1 ), 1, TAU( J ), WORK, N,
00339      $                      WORK( N**2+1 ) )
00340                V( J, J+1 ) = VSAVE
00341                WORK( ( N+1 )*J+1 ) = D( J+1 )
00342    60       CONTINUE
00343          END IF
00344 *
00345          DO 90 JCOL = 1, N
00346             IF( LOWER ) THEN
00347                DO 70 JROW = JCOL, N
00348                   WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) )
00349      $                - A( JROW, JCOL )
00350    70          CONTINUE
00351             ELSE
00352                DO 80 JROW = 1, JCOL
00353                   WORK( JROW+N*( JCOL-1 ) ) = WORK( JROW+N*( JCOL-1 ) )
00354      $                - A( JROW, JCOL )
00355    80          CONTINUE
00356             END IF
00357    90    CONTINUE
00358          WNORM = DLANSY( '1', CUPLO, N, WORK, N, WORK( N**2+1 ) )
00359 *
00360       ELSE IF( ITYPE.EQ.3 ) THEN
00361 *
00362 *        ITYPE=3: error = U V' - I
00363 *
00364          IF( N.LT.2 )
00365      $      RETURN
00366          CALL DLACPY( ' ', N, N, U, LDU, WORK, N )
00367          IF( LOWER ) THEN
00368             CALL DORM2R( 'R', 'T', N, N-1, N-1, V( 2, 1 ), LDV, TAU,
00369      $                   WORK( N+1 ), N, WORK( N**2+1 ), IINFO )
00370          ELSE
00371             CALL DORM2L( 'R', 'T', N, N-1, N-1, V( 1, 2 ), LDV, TAU,
00372      $                   WORK, N, WORK( N**2+1 ), IINFO )
00373          END IF
00374          IF( IINFO.NE.0 ) THEN
00375             RESULT( 1 ) = TEN / ULP
00376             RETURN
00377          END IF
00378 *
00379          DO 100 J = 1, N
00380             WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
00381   100    CONTINUE
00382 *
00383          WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) )
00384       END IF
00385 *
00386       IF( ANORM.GT.WNORM ) THEN
00387          RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
00388       ELSE
00389          IF( ANORM.LT.ONE ) THEN
00390             RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
00391          ELSE
00392             RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
00393          END IF
00394       END IF
00395 *
00396 *     Do Test 2
00397 *
00398 *     Compute  UU' - I
00399 *
00400       IF( ITYPE.EQ.1 ) THEN
00401          CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
00402      $               N )
00403 *
00404          DO 110 J = 1, N
00405             WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
00406   110    CONTINUE
00407 *
00408          RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N,
00409      $                 WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP )
00410       END IF
00411 *
00412       RETURN
00413 *
00414 *     End of DSYT21
00415 *
00416       END
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