LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spbtrf.f
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00001 *> \brief \b SPBTRF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download SPBTRF + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbtrf.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, KD, LDAB, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       REAL               AB( LDAB, * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> SPBTRF computes the Cholesky factorization of a real symmetric
00038 *> positive definite band matrix A.
00039 *>
00040 *> The factorization has the form
00041 *>    A = U**T * U,  if UPLO = 'U', or
00042 *>    A = L  * L**T,  if UPLO = 'L',
00043 *> where U is an upper triangular matrix and L is lower triangular.
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] UPLO
00050 *> \verbatim
00051 *>          UPLO is CHARACTER*1
00052 *>          = 'U':  Upper triangle of A is stored;
00053 *>          = 'L':  Lower triangle of A is stored.
00054 *> \endverbatim
00055 *>
00056 *> \param[in] N
00057 *> \verbatim
00058 *>          N is INTEGER
00059 *>          The order of the matrix A.  N >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] KD
00063 *> \verbatim
00064 *>          KD is INTEGER
00065 *>          The number of superdiagonals of the matrix A if UPLO = 'U',
00066 *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00067 *> \endverbatim
00068 *>
00069 *> \param[in,out] AB
00070 *> \verbatim
00071 *>          AB is REAL array, dimension (LDAB,N)
00072 *>          On entry, the upper or lower triangle of the symmetric band
00073 *>          matrix A, stored in the first KD+1 rows of the array.  The
00074 *>          j-th column of A is stored in the j-th column of the array AB
00075 *>          as follows:
00076 *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00077 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00078 *>
00079 *>          On exit, if INFO = 0, the triangular factor U or L from the
00080 *>          Cholesky factorization A = U**T*U or A = L*L**T of the band
00081 *>          matrix A, in the same storage format as A.
00082 *> \endverbatim
00083 *>
00084 *> \param[in] LDAB
00085 *> \verbatim
00086 *>          LDAB is INTEGER
00087 *>          The leading dimension of the array AB.  LDAB >= KD+1.
00088 *> \endverbatim
00089 *>
00090 *> \param[out] INFO
00091 *> \verbatim
00092 *>          INFO is INTEGER
00093 *>          = 0:  successful exit
00094 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00095 *>          > 0:  if INFO = i, the leading minor of order i is not
00096 *>                positive definite, and the factorization could not be
00097 *>                completed.
00098 *> \endverbatim
00099 *
00100 *  Authors:
00101 *  ========
00102 *
00103 *> \author Univ. of Tennessee 
00104 *> \author Univ. of California Berkeley 
00105 *> \author Univ. of Colorado Denver 
00106 *> \author NAG Ltd. 
00107 *
00108 *> \date November 2011
00109 *
00110 *> \ingroup realOTHERcomputational
00111 *
00112 *> \par Further Details:
00113 *  =====================
00114 *>
00115 *> \verbatim
00116 *>
00117 *>  The band storage scheme is illustrated by the following example, when
00118 *>  N = 6, KD = 2, and UPLO = 'U':
00119 *>
00120 *>  On entry:                       On exit:
00121 *>
00122 *>      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
00123 *>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
00124 *>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
00125 *>
00126 *>  Similarly, if UPLO = 'L' the format of A is as follows:
00127 *>
00128 *>  On entry:                       On exit:
00129 *>
00130 *>     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
00131 *>     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
00132 *>     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
00133 *>
00134 *>  Array elements marked * are not used by the routine.
00135 *> \endverbatim
00136 *
00137 *> \par Contributors:
00138 *  ==================
00139 *>
00140 *>  Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989
00141 *
00142 *  =====================================================================
00143       SUBROUTINE SPBTRF( UPLO, N, KD, AB, LDAB, INFO )
00144 *
00145 *  -- LAPACK computational routine (version 3.4.0) --
00146 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00147 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00148 *     November 2011
00149 *
00150 *     .. Scalar Arguments ..
00151       CHARACTER          UPLO
00152       INTEGER            INFO, KD, LDAB, N
00153 *     ..
00154 *     .. Array Arguments ..
00155       REAL               AB( LDAB, * )
00156 *     ..
00157 *
00158 *  =====================================================================
00159 *
00160 *     .. Parameters ..
00161       REAL               ONE, ZERO
00162       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00163       INTEGER            NBMAX, LDWORK
00164       PARAMETER          ( NBMAX = 32, LDWORK = NBMAX+1 )
00165 *     ..
00166 *     .. Local Scalars ..
00167       INTEGER            I, I2, I3, IB, II, J, JJ, NB
00168 *     ..
00169 *     .. Local Arrays ..
00170       REAL               WORK( LDWORK, NBMAX )
00171 *     ..
00172 *     .. External Functions ..
00173       LOGICAL            LSAME
00174       INTEGER            ILAENV
00175       EXTERNAL           LSAME, ILAENV
00176 *     ..
00177 *     .. External Subroutines ..
00178       EXTERNAL           SGEMM, SPBTF2, SPOTF2, SSYRK, STRSM, XERBLA
00179 *     ..
00180 *     .. Intrinsic Functions ..
00181       INTRINSIC          MIN
00182 *     ..
00183 *     .. Executable Statements ..
00184 *
00185 *     Test the input parameters.
00186 *
00187       INFO = 0
00188       IF( ( .NOT.LSAME( UPLO, 'U' ) ) .AND.
00189      $    ( .NOT.LSAME( UPLO, 'L' ) ) ) THEN
00190          INFO = -1
00191       ELSE IF( N.LT.0 ) THEN
00192          INFO = -2
00193       ELSE IF( KD.LT.0 ) THEN
00194          INFO = -3
00195       ELSE IF( LDAB.LT.KD+1 ) THEN
00196          INFO = -5
00197       END IF
00198       IF( INFO.NE.0 ) THEN
00199          CALL XERBLA( 'SPBTRF', -INFO )
00200          RETURN
00201       END IF
00202 *
00203 *     Quick return if possible
00204 *
00205       IF( N.EQ.0 )
00206      $   RETURN
00207 *
00208 *     Determine the block size for this environment
00209 *
00210       NB = ILAENV( 1, 'SPBTRF', UPLO, N, KD, -1, -1 )
00211 *
00212 *     The block size must not exceed the semi-bandwidth KD, and must not
00213 *     exceed the limit set by the size of the local array WORK.
00214 *
00215       NB = MIN( NB, NBMAX )
00216 *
00217       IF( NB.LE.1 .OR. NB.GT.KD ) THEN
00218 *
00219 *        Use unblocked code
00220 *
00221          CALL SPBTF2( UPLO, N, KD, AB, LDAB, INFO )
00222       ELSE
00223 *
00224 *        Use blocked code
00225 *
00226          IF( LSAME( UPLO, 'U' ) ) THEN
00227 *
00228 *           Compute the Cholesky factorization of a symmetric band
00229 *           matrix, given the upper triangle of the matrix in band
00230 *           storage.
00231 *
00232 *           Zero the upper triangle of the work array.
00233 *
00234             DO 20 J = 1, NB
00235                DO 10 I = 1, J - 1
00236                   WORK( I, J ) = ZERO
00237    10          CONTINUE
00238    20       CONTINUE
00239 *
00240 *           Process the band matrix one diagonal block at a time.
00241 *
00242             DO 70 I = 1, N, NB
00243                IB = MIN( NB, N-I+1 )
00244 *
00245 *              Factorize the diagonal block
00246 *
00247                CALL SPOTF2( UPLO, IB, AB( KD+1, I ), LDAB-1, II )
00248                IF( II.NE.0 ) THEN
00249                   INFO = I + II - 1
00250                   GO TO 150
00251                END IF
00252                IF( I+IB.LE.N ) THEN
00253 *
00254 *                 Update the relevant part of the trailing submatrix.
00255 *                 If A11 denotes the diagonal block which has just been
00256 *                 factorized, then we need to update the remaining
00257 *                 blocks in the diagram:
00258 *
00259 *                    A11   A12   A13
00260 *                          A22   A23
00261 *                                A33
00262 *
00263 *                 The numbers of rows and columns in the partitioning
00264 *                 are IB, I2, I3 respectively. The blocks A12, A22 and
00265 *                 A23 are empty if IB = KD. The upper triangle of A13
00266 *                 lies outside the band.
00267 *
00268                   I2 = MIN( KD-IB, N-I-IB+1 )
00269                   I3 = MIN( IB, N-I-KD+1 )
00270 *
00271                   IF( I2.GT.0 ) THEN
00272 *
00273 *                    Update A12
00274 *
00275                      CALL STRSM( 'Left', 'Upper', 'Transpose',
00276      $                           'Non-unit', IB, I2, ONE, AB( KD+1, I ),
00277      $                           LDAB-1, AB( KD+1-IB, I+IB ), LDAB-1 )
00278 *
00279 *                    Update A22
00280 *
00281                      CALL SSYRK( 'Upper', 'Transpose', I2, IB, -ONE,
00282      $                           AB( KD+1-IB, I+IB ), LDAB-1, ONE,
00283      $                           AB( KD+1, I+IB ), LDAB-1 )
00284                   END IF
00285 *
00286                   IF( I3.GT.0 ) THEN
00287 *
00288 *                    Copy the lower triangle of A13 into the work array.
00289 *
00290                      DO 40 JJ = 1, I3
00291                         DO 30 II = JJ, IB
00292                            WORK( II, JJ ) = AB( II-JJ+1, JJ+I+KD-1 )
00293    30                   CONTINUE
00294    40                CONTINUE
00295 *
00296 *                    Update A13 (in the work array).
00297 *
00298                      CALL STRSM( 'Left', 'Upper', 'Transpose',
00299      $                           'Non-unit', IB, I3, ONE, AB( KD+1, I ),
00300      $                           LDAB-1, WORK, LDWORK )
00301 *
00302 *                    Update A23
00303 *
00304                      IF( I2.GT.0 )
00305      $                  CALL SGEMM( 'Transpose', 'No Transpose', I2, I3,
00306      $                              IB, -ONE, AB( KD+1-IB, I+IB ),
00307      $                              LDAB-1, WORK, LDWORK, ONE,
00308      $                              AB( 1+IB, I+KD ), LDAB-1 )
00309 *
00310 *                    Update A33
00311 *
00312                      CALL SSYRK( 'Upper', 'Transpose', I3, IB, -ONE,
00313      $                           WORK, LDWORK, ONE, AB( KD+1, I+KD ),
00314      $                           LDAB-1 )
00315 *
00316 *                    Copy the lower triangle of A13 back into place.
00317 *
00318                      DO 60 JJ = 1, I3
00319                         DO 50 II = JJ, IB
00320                            AB( II-JJ+1, JJ+I+KD-1 ) = WORK( II, JJ )
00321    50                   CONTINUE
00322    60                CONTINUE
00323                   END IF
00324                END IF
00325    70       CONTINUE
00326          ELSE
00327 *
00328 *           Compute the Cholesky factorization of a symmetric band
00329 *           matrix, given the lower triangle of the matrix in band
00330 *           storage.
00331 *
00332 *           Zero the lower triangle of the work array.
00333 *
00334             DO 90 J = 1, NB
00335                DO 80 I = J + 1, NB
00336                   WORK( I, J ) = ZERO
00337    80          CONTINUE
00338    90       CONTINUE
00339 *
00340 *           Process the band matrix one diagonal block at a time.
00341 *
00342             DO 140 I = 1, N, NB
00343                IB = MIN( NB, N-I+1 )
00344 *
00345 *              Factorize the diagonal block
00346 *
00347                CALL SPOTF2( UPLO, IB, AB( 1, I ), LDAB-1, II )
00348                IF( II.NE.0 ) THEN
00349                   INFO = I + II - 1
00350                   GO TO 150
00351                END IF
00352                IF( I+IB.LE.N ) THEN
00353 *
00354 *                 Update the relevant part of the trailing submatrix.
00355 *                 If A11 denotes the diagonal block which has just been
00356 *                 factorized, then we need to update the remaining
00357 *                 blocks in the diagram:
00358 *
00359 *                    A11
00360 *                    A21   A22
00361 *                    A31   A32   A33
00362 *
00363 *                 The numbers of rows and columns in the partitioning
00364 *                 are IB, I2, I3 respectively. The blocks A21, A22 and
00365 *                 A32 are empty if IB = KD. The lower triangle of A31
00366 *                 lies outside the band.
00367 *
00368                   I2 = MIN( KD-IB, N-I-IB+1 )
00369                   I3 = MIN( IB, N-I-KD+1 )
00370 *
00371                   IF( I2.GT.0 ) THEN
00372 *
00373 *                    Update A21
00374 *
00375                      CALL STRSM( 'Right', 'Lower', 'Transpose',
00376      $                           'Non-unit', I2, IB, ONE, AB( 1, I ),
00377      $                           LDAB-1, AB( 1+IB, I ), LDAB-1 )
00378 *
00379 *                    Update A22
00380 *
00381                      CALL SSYRK( 'Lower', 'No Transpose', I2, IB, -ONE,
00382      $                           AB( 1+IB, I ), LDAB-1, ONE,
00383      $                           AB( 1, I+IB ), LDAB-1 )
00384                   END IF
00385 *
00386                   IF( I3.GT.0 ) THEN
00387 *
00388 *                    Copy the upper triangle of A31 into the work array.
00389 *
00390                      DO 110 JJ = 1, IB
00391                         DO 100 II = 1, MIN( JJ, I3 )
00392                            WORK( II, JJ ) = AB( KD+1-JJ+II, JJ+I-1 )
00393   100                   CONTINUE
00394   110                CONTINUE
00395 *
00396 *                    Update A31 (in the work array).
00397 *
00398                      CALL STRSM( 'Right', 'Lower', 'Transpose',
00399      $                           'Non-unit', I3, IB, ONE, AB( 1, I ),
00400      $                           LDAB-1, WORK, LDWORK )
00401 *
00402 *                    Update A32
00403 *
00404                      IF( I2.GT.0 )
00405      $                  CALL SGEMM( 'No transpose', 'Transpose', I3, I2,
00406      $                              IB, -ONE, WORK, LDWORK,
00407      $                              AB( 1+IB, I ), LDAB-1, ONE,
00408      $                              AB( 1+KD-IB, I+IB ), LDAB-1 )
00409 *
00410 *                    Update A33
00411 *
00412                      CALL SSYRK( 'Lower', 'No Transpose', I3, IB, -ONE,
00413      $                           WORK, LDWORK, ONE, AB( 1, I+KD ),
00414      $                           LDAB-1 )
00415 *
00416 *                    Copy the upper triangle of A31 back into place.
00417 *
00418                      DO 130 JJ = 1, IB
00419                         DO 120 II = 1, MIN( JJ, I3 )
00420                            AB( KD+1-JJ+II, JJ+I-1 ) = WORK( II, JJ )
00421   120                   CONTINUE
00422   130                CONTINUE
00423                   END IF
00424                END IF
00425   140       CONTINUE
00426          END IF
00427       END IF
00428       RETURN
00429 *
00430   150 CONTINUE
00431       RETURN
00432 *
00433 *     End of SPBTRF
00434 *
00435       END
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