LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dspt21.f
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00001 *> \brief \b DSPT21
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 DSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
00012 *                          TAU, WORK, RESULT )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            ITYPE, KBAND, LDU, N
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       DOUBLE PRECISION   AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
00020 *      $                   U( LDU, * ), VP( * ), WORK( * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> DSPT21  generally checks a decomposition of the form
00030 *>
00031 *>         A = U S U'
00032 *>
00033 *> where ' means transpose, A is symmetric (stored in packed format), U
00034 *> is orthogonal, and S is diagonal (if KBAND=0) or symmetric
00035 *> tridiagonal (if KBAND=1).  If ITYPE=1, then U is represented as a
00036 *> dense matrix, otherwise the U is expressed as a product of
00037 *> Householder transformations, whose vectors are stored in the array
00038 *> "V" and whose scaling constants are in "TAU"; we shall use the
00039 *> letter "V" to refer to the product of Householder transformations
00040 *> (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 *> Packed storage means that, for example, if UPLO='U', then the columns
00055 *> of the upper triangle of A are stored one after another, so that
00056 *> A(1,j+1) immediately follows A(j,j) in the array AP.  Similarly, if
00057 *> UPLO='L', then the columns of the lower triangle of A are stored one
00058 *> after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
00059 *> in the array AP.  This means that A(i,j) is stored in:
00060 *>
00061 *>    AP( i + j*(j-1)/2 )                 if UPLO='U'
00062 *>
00063 *>    AP( i + (2*n-j)*(j-1)/2 )           if UPLO='L'
00064 *>
00065 *> The array VP bears the same relation to the matrix V that A does to
00066 *> AP.
00067 *>
00068 *> For ITYPE > 1, the transformation U is expressed as a product
00069 *> of Householder transformations:
00070 *>
00071 *>    If UPLO='U', then  V = H(n-1)...H(1),  where
00072 *>
00073 *>        H(j) = I  -  tau(j) v(j) v(j)'
00074 *>
00075 *>    and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
00076 *>    (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
00077 *>    the j-th element is 1, and the last n-j elements are 0.
00078 *>
00079 *>    If UPLO='L', then  V = H(1)...H(n-1),  where
00080 *>
00081 *>        H(j) = I  -  tau(j) v(j) v(j)'
00082 *>
00083 *>    and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
00084 *>    (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
00085 *>    in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
00086 *> \endverbatim
00087 *
00088 *  Arguments:
00089 *  ==========
00090 *
00091 *> \param[in] ITYPE
00092 *> \verbatim
00093 *>          ITYPE is INTEGER
00094 *>          Specifies the type of tests to be performed.
00095 *>          1: U expressed as a dense orthogonal matrix:
00096 *>             RESULT(1) = | A - U S U' | / ( |A| n ulp )   *andC>             RESULT(2) = | I - UU' | / ( n ulp )
00097 *>
00098 *>          2: U expressed as a product V of Housholder transformations:
00099 *>             RESULT(1) = | A - V S V' | / ( |A| n ulp )
00100 *>
00101 *>          3: U expressed both as a dense orthogonal matrix and
00102 *>             as a product of Housholder transformations:
00103 *>             RESULT(1) = | I - VU' | / ( n ulp )
00104 *> \endverbatim
00105 *>
00106 *> \param[in] UPLO
00107 *> \verbatim
00108 *>          UPLO is CHARACTER
00109 *>          If UPLO='U', AP and VP are considered to contain the upper
00110 *>          triangle of A and V.
00111 *>          If UPLO='L', AP and VP are considered to contain the lower
00112 *>          triangle of A and V.
00113 *> \endverbatim
00114 *>
00115 *> \param[in] N
00116 *> \verbatim
00117 *>          N is INTEGER
00118 *>          The size of the matrix.  If it is zero, DSPT21 does nothing.
00119 *>          It must be at least zero.
00120 *> \endverbatim
00121 *>
00122 *> \param[in] KBAND
00123 *> \verbatim
00124 *>          KBAND is INTEGER
00125 *>          The bandwidth of the matrix.  It may only be zero or one.
00126 *>          If zero, then S is diagonal, and E is not referenced.  If
00127 *>          one, then S is symmetric tri-diagonal.
00128 *> \endverbatim
00129 *>
00130 *> \param[in] AP
00131 *> \verbatim
00132 *>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
00133 *>          The original (unfactored) matrix.  It is assumed to be
00134 *>          symmetric, and contains the columns of just the upper
00135 *>          triangle (UPLO='U') or only the lower triangle (UPLO='L'),
00136 *>          packed one after another.
00137 *> \endverbatim
00138 *>
00139 *> \param[in] D
00140 *> \verbatim
00141 *>          D is DOUBLE PRECISION array, dimension (N)
00142 *>          The diagonal of the (symmetric tri-) diagonal matrix.
00143 *> \endverbatim
00144 *>
00145 *> \param[in] E
00146 *> \verbatim
00147 *>          E is DOUBLE PRECISION array, dimension (N-1)
00148 *>          The off-diagonal of the (symmetric tri-) diagonal matrix.
00149 *>          E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
00150 *>          (3,2) element, etc.
00151 *>          Not referenced if KBAND=0.
00152 *> \endverbatim
00153 *>
00154 *> \param[in] U
00155 *> \verbatim
00156 *>          U is DOUBLE PRECISION array, dimension (LDU, N)
00157 *>          If ITYPE=1 or 3, this contains the orthogonal matrix in
00158 *>          the decomposition, expressed as a dense matrix.  If ITYPE=2,
00159 *>          then it is not referenced.
00160 *> \endverbatim
00161 *>
00162 *> \param[in] LDU
00163 *> \verbatim
00164 *>          LDU is INTEGER
00165 *>          The leading dimension of U.  LDU must be at least N and
00166 *>          at least 1.
00167 *> \endverbatim
00168 *>
00169 *> \param[in] VP
00170 *> \verbatim
00171 *>          VP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
00172 *>          If ITYPE=2 or 3, the columns of this array contain the
00173 *>          Householder vectors used to describe the orthogonal matrix
00174 *>          in the decomposition, as described in purpose.
00175 *>          *NOTE* If ITYPE=2 or 3, V is modified and restored.  The
00176 *>          subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
00177 *>          is set to one, and later reset to its original value, during
00178 *>          the course of the calculation.
00179 *>          If ITYPE=1, then it is neither referenced nor modified.
00180 *> \endverbatim
00181 *>
00182 *> \param[in] TAU
00183 *> \verbatim
00184 *>          TAU is DOUBLE PRECISION array, dimension (N)
00185 *>          If ITYPE >= 2, then TAU(j) is the scalar factor of
00186 *>          v(j) v(j)' in the Householder transformation H(j) of
00187 *>          the product  U = H(1)...H(n-2)
00188 *>          If ITYPE < 2, then TAU is not referenced.
00189 *> \endverbatim
00190 *>
00191 *> \param[out] WORK
00192 *> \verbatim
00193 *>          WORK is DOUBLE PRECISION array, dimension (N**2+N)
00194 *>          Workspace.
00195 *> \endverbatim
00196 *>
00197 *> \param[out] RESULT
00198 *> \verbatim
00199 *>          RESULT is DOUBLE PRECISION array, dimension (2)
00200 *>          The values computed by the two tests described above.  The
00201 *>          values are currently limited to 1/ulp, to avoid overflow.
00202 *>          RESULT(1) is always modified.  RESULT(2) is modified only
00203 *>          if ITYPE=1.
00204 *> \endverbatim
00205 *
00206 *  Authors:
00207 *  ========
00208 *
00209 *> \author Univ. of Tennessee 
00210 *> \author Univ. of California Berkeley 
00211 *> \author Univ. of Colorado Denver 
00212 *> \author NAG Ltd. 
00213 *
00214 *> \date November 2011
00215 *
00216 *> \ingroup double_eig
00217 *
00218 *  =====================================================================
00219       SUBROUTINE DSPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
00220      $                   TAU, WORK, RESULT )
00221 *
00222 *  -- LAPACK test routine (version 3.4.0) --
00223 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00224 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00225 *     November 2011
00226 *
00227 *     .. Scalar Arguments ..
00228       CHARACTER          UPLO
00229       INTEGER            ITYPE, KBAND, LDU, N
00230 *     ..
00231 *     .. Array Arguments ..
00232       DOUBLE PRECISION   AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
00233      $                   U( LDU, * ), VP( * ), WORK( * )
00234 *     ..
00235 *
00236 *  =====================================================================
00237 *
00238 *     .. Parameters ..
00239       DOUBLE PRECISION   ZERO, ONE, TEN
00240       PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TEN = 10.0D0 )
00241       DOUBLE PRECISION   HALF
00242       PARAMETER          ( HALF = 1.0D+0 / 2.0D+0 )
00243 *     ..
00244 *     .. Local Scalars ..
00245       LOGICAL            LOWER
00246       CHARACTER          CUPLO
00247       INTEGER            IINFO, J, JP, JP1, JR, LAP
00248       DOUBLE PRECISION   ANORM, TEMP, ULP, UNFL, VSAVE, WNORM
00249 *     ..
00250 *     .. External Functions ..
00251       LOGICAL            LSAME
00252       DOUBLE PRECISION   DDOT, DLAMCH, DLANGE, DLANSP
00253       EXTERNAL           LSAME, DDOT, DLAMCH, DLANGE, DLANSP
00254 *     ..
00255 *     .. External Subroutines ..
00256       EXTERNAL           DAXPY, DCOPY, DGEMM, DLACPY, DLASET, DOPMTR,
00257      $                   DSPMV, DSPR, DSPR2
00258 *     ..
00259 *     .. Intrinsic Functions ..
00260       INTRINSIC          DBLE, MAX, MIN
00261 *     ..
00262 *     .. Executable Statements ..
00263 *
00264 *     1)      Constants
00265 *
00266       RESULT( 1 ) = ZERO
00267       IF( ITYPE.EQ.1 )
00268      $   RESULT( 2 ) = ZERO
00269       IF( N.LE.0 )
00270      $   RETURN
00271 *
00272       LAP = ( N*( N+1 ) ) / 2
00273 *
00274       IF( LSAME( UPLO, 'U' ) ) THEN
00275          LOWER = .FALSE.
00276          CUPLO = 'U'
00277       ELSE
00278          LOWER = .TRUE.
00279          CUPLO = 'L'
00280       END IF
00281 *
00282       UNFL = DLAMCH( 'Safe minimum' )
00283       ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
00284 *
00285 *     Some Error Checks
00286 *
00287       IF( ITYPE.LT.1 .OR. ITYPE.GT.3 ) THEN
00288          RESULT( 1 ) = TEN / ULP
00289          RETURN
00290       END IF
00291 *
00292 *     Do Test 1
00293 *
00294 *     Norm of A:
00295 *
00296       IF( ITYPE.EQ.3 ) THEN
00297          ANORM = ONE
00298       ELSE
00299          ANORM = MAX( DLANSP( '1', CUPLO, N, AP, WORK ), UNFL )
00300       END IF
00301 *
00302 *     Compute error matrix:
00303 *
00304       IF( ITYPE.EQ.1 ) THEN
00305 *
00306 *        ITYPE=1: error = A - U S U'
00307 *
00308          CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
00309          CALL DCOPY( LAP, AP, 1, WORK, 1 )
00310 *
00311          DO 10 J = 1, N
00312             CALL DSPR( CUPLO, N, -D( J ), U( 1, J ), 1, WORK )
00313    10    CONTINUE
00314 *
00315          IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
00316             DO 20 J = 1, N - 1
00317                CALL DSPR2( CUPLO, N, -E( J ), U( 1, J ), 1, U( 1, J+1 ),
00318      $                     1, WORK )
00319    20       CONTINUE
00320          END IF
00321          WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( N**2+1 ) )
00322 *
00323       ELSE IF( ITYPE.EQ.2 ) THEN
00324 *
00325 *        ITYPE=2: error = V S V' - A
00326 *
00327          CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
00328 *
00329          IF( LOWER ) THEN
00330             WORK( LAP ) = D( N )
00331             DO 40 J = N - 1, 1, -1
00332                JP = ( ( 2*N-J )*( J-1 ) ) / 2
00333                JP1 = JP + N - J
00334                IF( KBAND.EQ.1 ) THEN
00335                   WORK( JP+J+1 ) = ( ONE-TAU( J ) )*E( J )
00336                   DO 30 JR = J + 2, N
00337                      WORK( JP+JR ) = -TAU( J )*E( J )*VP( JP+JR )
00338    30             CONTINUE
00339                END IF
00340 *
00341                IF( TAU( J ).NE.ZERO ) THEN
00342                   VSAVE = VP( JP+J+1 )
00343                   VP( JP+J+1 ) = ONE
00344                   CALL DSPMV( 'L', N-J, ONE, WORK( JP1+J+1 ),
00345      $                        VP( JP+J+1 ), 1, ZERO, WORK( LAP+1 ), 1 )
00346                   TEMP = -HALF*TAU( J )*DDOT( N-J, WORK( LAP+1 ), 1,
00347      $                   VP( JP+J+1 ), 1 )
00348                   CALL DAXPY( N-J, TEMP, VP( JP+J+1 ), 1, WORK( LAP+1 ),
00349      $                        1 )
00350                   CALL DSPR2( 'L', N-J, -TAU( J ), VP( JP+J+1 ), 1,
00351      $                        WORK( LAP+1 ), 1, WORK( JP1+J+1 ) )
00352                   VP( JP+J+1 ) = VSAVE
00353                END IF
00354                WORK( JP+J ) = D( J )
00355    40       CONTINUE
00356          ELSE
00357             WORK( 1 ) = D( 1 )
00358             DO 60 J = 1, N - 1
00359                JP = ( J*( J-1 ) ) / 2
00360                JP1 = JP + J
00361                IF( KBAND.EQ.1 ) THEN
00362                   WORK( JP1+J ) = ( ONE-TAU( J ) )*E( J )
00363                   DO 50 JR = 1, J - 1
00364                      WORK( JP1+JR ) = -TAU( J )*E( J )*VP( JP1+JR )
00365    50             CONTINUE
00366                END IF
00367 *
00368                IF( TAU( J ).NE.ZERO ) THEN
00369                   VSAVE = VP( JP1+J )
00370                   VP( JP1+J ) = ONE
00371                   CALL DSPMV( 'U', J, ONE, WORK, VP( JP1+1 ), 1, ZERO,
00372      $                        WORK( LAP+1 ), 1 )
00373                   TEMP = -HALF*TAU( J )*DDOT( J, WORK( LAP+1 ), 1,
00374      $                   VP( JP1+1 ), 1 )
00375                   CALL DAXPY( J, TEMP, VP( JP1+1 ), 1, WORK( LAP+1 ),
00376      $                        1 )
00377                   CALL DSPR2( 'U', J, -TAU( J ), VP( JP1+1 ), 1,
00378      $                        WORK( LAP+1 ), 1, WORK )
00379                   VP( JP1+J ) = VSAVE
00380                END IF
00381                WORK( JP1+J+1 ) = D( J+1 )
00382    60       CONTINUE
00383          END IF
00384 *
00385          DO 70 J = 1, LAP
00386             WORK( J ) = WORK( J ) - AP( J )
00387    70    CONTINUE
00388          WNORM = DLANSP( '1', CUPLO, N, WORK, WORK( LAP+1 ) )
00389 *
00390       ELSE IF( ITYPE.EQ.3 ) THEN
00391 *
00392 *        ITYPE=3: error = U V' - I
00393 *
00394          IF( N.LT.2 )
00395      $      RETURN
00396          CALL DLACPY( ' ', N, N, U, LDU, WORK, N )
00397          CALL DOPMTR( 'R', CUPLO, 'T', N, N, VP, TAU, WORK, N,
00398      $                WORK( N**2+1 ), IINFO )
00399          IF( IINFO.NE.0 ) THEN
00400             RESULT( 1 ) = TEN / ULP
00401             RETURN
00402          END IF
00403 *
00404          DO 80 J = 1, N
00405             WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
00406    80    CONTINUE
00407 *
00408          WNORM = DLANGE( '1', N, N, WORK, N, WORK( N**2+1 ) )
00409       END IF
00410 *
00411       IF( ANORM.GT.WNORM ) THEN
00412          RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
00413       ELSE
00414          IF( ANORM.LT.ONE ) THEN
00415             RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
00416          ELSE
00417             RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
00418          END IF
00419       END IF
00420 *
00421 *     Do Test 2
00422 *
00423 *     Compute  UU' - I
00424 *
00425       IF( ITYPE.EQ.1 ) THEN
00426          CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
00427      $               N )
00428 *
00429          DO 90 J = 1, N
00430             WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
00431    90    CONTINUE
00432 *
00433          RESULT( 2 ) = MIN( DLANGE( '1', N, N, WORK, N,
00434      $                 WORK( N**2+1 ) ), DBLE( N ) ) / ( N*ULP )
00435       END IF
00436 *
00437       RETURN
00438 *
00439 *     End of DSPT21
00440 *
00441       END
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