LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zgbtrf.f
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00001 *> \brief \b ZGBTRF
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 ZGBTRF + 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/zgbtrf.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       INTEGER            INFO, KL, KU, LDAB, M, N
00025 *       ..
00026 *       .. Array Arguments ..
00027 *       INTEGER            IPIV( * )
00028 *       COMPLEX*16         AB( LDAB, * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> ZGBTRF computes an LU factorization of a complex m-by-n band matrix A
00038 *> using partial pivoting with row interchanges.
00039 *>
00040 *> This is the blocked version of the algorithm, calling Level 3 BLAS.
00041 *> \endverbatim
00042 *
00043 *  Arguments:
00044 *  ==========
00045 *
00046 *> \param[in] M
00047 *> \verbatim
00048 *>          M is INTEGER
00049 *>          The number of rows of the matrix A.  M >= 0.
00050 *> \endverbatim
00051 *>
00052 *> \param[in] N
00053 *> \verbatim
00054 *>          N is INTEGER
00055 *>          The number of columns of the matrix A.  N >= 0.
00056 *> \endverbatim
00057 *>
00058 *> \param[in] KL
00059 *> \verbatim
00060 *>          KL is INTEGER
00061 *>          The number of subdiagonals within the band of A.  KL >= 0.
00062 *> \endverbatim
00063 *>
00064 *> \param[in] KU
00065 *> \verbatim
00066 *>          KU is INTEGER
00067 *>          The number of superdiagonals within the band of A.  KU >= 0.
00068 *> \endverbatim
00069 *>
00070 *> \param[in,out] AB
00071 *> \verbatim
00072 *>          AB is COMPLEX*16 array, dimension (LDAB,N)
00073 *>          On entry, the matrix A in band storage, in rows KL+1 to
00074 *>          2*KL+KU+1; rows 1 to KL of the array need not be set.
00075 *>          The j-th column of A is stored in the j-th column of the
00076 *>          array AB as follows:
00077 *>          AB(kl+ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl)
00078 *>
00079 *>          On exit, details of the factorization: U is stored as an
00080 *>          upper triangular band matrix with KL+KU superdiagonals in
00081 *>          rows 1 to KL+KU+1, and the multipliers used during the
00082 *>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
00083 *>          See below for further details.
00084 *> \endverbatim
00085 *>
00086 *> \param[in] LDAB
00087 *> \verbatim
00088 *>          LDAB is INTEGER
00089 *>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
00090 *> \endverbatim
00091 *>
00092 *> \param[out] IPIV
00093 *> \verbatim
00094 *>          IPIV is INTEGER array, dimension (min(M,N))
00095 *>          The pivot indices; for 1 <= i <= min(M,N), row i of the
00096 *>          matrix was interchanged with row IPIV(i).
00097 *> \endverbatim
00098 *>
00099 *> \param[out] INFO
00100 *> \verbatim
00101 *>          INFO is INTEGER
00102 *>          = 0: successful exit
00103 *>          < 0: if INFO = -i, the i-th argument had an illegal value
00104 *>          > 0: if INFO = +i, U(i,i) is exactly zero. The factorization
00105 *>               has been completed, but the factor U is exactly
00106 *>               singular, and division by zero will occur if it is used
00107 *>               to solve a system of equations.
00108 *> \endverbatim
00109 *
00110 *  Authors:
00111 *  ========
00112 *
00113 *> \author Univ. of Tennessee 
00114 *> \author Univ. of California Berkeley 
00115 *> \author Univ. of Colorado Denver 
00116 *> \author NAG Ltd. 
00117 *
00118 *> \date November 2011
00119 *
00120 *> \ingroup complex16GBcomputational
00121 *
00122 *> \par Further Details:
00123 *  =====================
00124 *>
00125 *> \verbatim
00126 *>
00127 *>  The band storage scheme is illustrated by the following example, when
00128 *>  M = N = 6, KL = 2, KU = 1:
00129 *>
00130 *>  On entry:                       On exit:
00131 *>
00132 *>      *    *    *    +    +    +       *    *    *   u14  u25  u36
00133 *>      *    *    +    +    +    +       *    *   u13  u24  u35  u46
00134 *>      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
00135 *>     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
00136 *>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
00137 *>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
00138 *>
00139 *>  Array elements marked * are not used by the routine; elements marked
00140 *>  + need not be set on entry, but are required by the routine to store
00141 *>  elements of U because of fill-in resulting from the row interchanges.
00142 *> \endverbatim
00143 *>
00144 *  =====================================================================
00145       SUBROUTINE ZGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
00146 *
00147 *  -- LAPACK computational routine (version 3.4.0) --
00148 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00149 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00150 *     November 2011
00151 *
00152 *     .. Scalar Arguments ..
00153       INTEGER            INFO, KL, KU, LDAB, M, N
00154 *     ..
00155 *     .. Array Arguments ..
00156       INTEGER            IPIV( * )
00157       COMPLEX*16         AB( LDAB, * )
00158 *     ..
00159 *
00160 *  =====================================================================
00161 *
00162 *     .. Parameters ..
00163       COMPLEX*16         ONE, ZERO
00164       PARAMETER          ( ONE = ( 1.0D+0, 0.0D+0 ),
00165      $                   ZERO = ( 0.0D+0, 0.0D+0 ) )
00166       INTEGER            NBMAX, LDWORK
00167       PARAMETER          ( NBMAX = 64, LDWORK = NBMAX+1 )
00168 *     ..
00169 *     .. Local Scalars ..
00170       INTEGER            I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
00171      $                   JU, K2, KM, KV, NB, NW
00172       COMPLEX*16         TEMP
00173 *     ..
00174 *     .. Local Arrays ..
00175       COMPLEX*16         WORK13( LDWORK, NBMAX ),
00176      $                   WORK31( LDWORK, NBMAX )
00177 *     ..
00178 *     .. External Functions ..
00179       INTEGER            ILAENV, IZAMAX
00180       EXTERNAL           ILAENV, IZAMAX
00181 *     ..
00182 *     .. External Subroutines ..
00183       EXTERNAL           XERBLA, ZCOPY, ZGBTF2, ZGEMM, ZGERU, ZLASWP,
00184      $                   ZSCAL, ZSWAP, ZTRSM
00185 *     ..
00186 *     .. Intrinsic Functions ..
00187       INTRINSIC          MAX, MIN
00188 *     ..
00189 *     .. Executable Statements ..
00190 *
00191 *     KV is the number of superdiagonals in the factor U, allowing for
00192 *     fill-in
00193 *
00194       KV = KU + KL
00195 *
00196 *     Test the input parameters.
00197 *
00198       INFO = 0
00199       IF( M.LT.0 ) THEN
00200          INFO = -1
00201       ELSE IF( N.LT.0 ) THEN
00202          INFO = -2
00203       ELSE IF( KL.LT.0 ) THEN
00204          INFO = -3
00205       ELSE IF( KU.LT.0 ) THEN
00206          INFO = -4
00207       ELSE IF( LDAB.LT.KL+KV+1 ) THEN
00208          INFO = -6
00209       END IF
00210       IF( INFO.NE.0 ) THEN
00211          CALL XERBLA( 'ZGBTRF', -INFO )
00212          RETURN
00213       END IF
00214 *
00215 *     Quick return if possible
00216 *
00217       IF( M.EQ.0 .OR. N.EQ.0 )
00218      $   RETURN
00219 *
00220 *     Determine the block size for this environment
00221 *
00222       NB = ILAENV( 1, 'ZGBTRF', ' ', M, N, KL, KU )
00223 *
00224 *     The block size must not exceed the limit set by the size of the
00225 *     local arrays WORK13 and WORK31.
00226 *
00227       NB = MIN( NB, NBMAX )
00228 *
00229       IF( NB.LE.1 .OR. NB.GT.KL ) THEN
00230 *
00231 *        Use unblocked code
00232 *
00233          CALL ZGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
00234       ELSE
00235 *
00236 *        Use blocked code
00237 *
00238 *        Zero the superdiagonal elements of the work array WORK13
00239 *
00240          DO 20 J = 1, NB
00241             DO 10 I = 1, J - 1
00242                WORK13( I, J ) = ZERO
00243    10       CONTINUE
00244    20    CONTINUE
00245 *
00246 *        Zero the subdiagonal elements of the work array WORK31
00247 *
00248          DO 40 J = 1, NB
00249             DO 30 I = J + 1, NB
00250                WORK31( I, J ) = ZERO
00251    30       CONTINUE
00252    40    CONTINUE
00253 *
00254 *        Gaussian elimination with partial pivoting
00255 *
00256 *        Set fill-in elements in columns KU+2 to KV to zero
00257 *
00258          DO 60 J = KU + 2, MIN( KV, N )
00259             DO 50 I = KV - J + 2, KL
00260                AB( I, J ) = ZERO
00261    50       CONTINUE
00262    60    CONTINUE
00263 *
00264 *        JU is the index of the last column affected by the current
00265 *        stage of the factorization
00266 *
00267          JU = 1
00268 *
00269          DO 180 J = 1, MIN( M, N ), NB
00270             JB = MIN( NB, MIN( M, N )-J+1 )
00271 *
00272 *           The active part of the matrix is partitioned
00273 *
00274 *              A11   A12   A13
00275 *              A21   A22   A23
00276 *              A31   A32   A33
00277 *
00278 *           Here A11, A21 and A31 denote the current block of JB columns
00279 *           which is about to be factorized. The number of rows in the
00280 *           partitioning are JB, I2, I3 respectively, and the numbers
00281 *           of columns are JB, J2, J3. The superdiagonal elements of A13
00282 *           and the subdiagonal elements of A31 lie outside the band.
00283 *
00284             I2 = MIN( KL-JB, M-J-JB+1 )
00285             I3 = MIN( JB, M-J-KL+1 )
00286 *
00287 *           J2 and J3 are computed after JU has been updated.
00288 *
00289 *           Factorize the current block of JB columns
00290 *
00291             DO 80 JJ = J, J + JB - 1
00292 *
00293 *              Set fill-in elements in column JJ+KV to zero
00294 *
00295                IF( JJ+KV.LE.N ) THEN
00296                   DO 70 I = 1, KL
00297                      AB( I, JJ+KV ) = ZERO
00298    70             CONTINUE
00299                END IF
00300 *
00301 *              Find pivot and test for singularity. KM is the number of
00302 *              subdiagonal elements in the current column.
00303 *
00304                KM = MIN( KL, M-JJ )
00305                JP = IZAMAX( KM+1, AB( KV+1, JJ ), 1 )
00306                IPIV( JJ ) = JP + JJ - J
00307                IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
00308                   JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
00309                   IF( JP.NE.1 ) THEN
00310 *
00311 *                    Apply interchange to columns J to J+JB-1
00312 *
00313                      IF( JP+JJ-1.LT.J+KL ) THEN
00314 *
00315                         CALL ZSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
00316      $                              AB( KV+JP+JJ-J, J ), LDAB-1 )
00317                      ELSE
00318 *
00319 *                       The interchange affects columns J to JJ-1 of A31
00320 *                       which are stored in the work array WORK31
00321 *
00322                         CALL ZSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
00323      $                              WORK31( JP+JJ-J-KL, 1 ), LDWORK )
00324                         CALL ZSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
00325      $                              AB( KV+JP, JJ ), LDAB-1 )
00326                      END IF
00327                   END IF
00328 *
00329 *                 Compute multipliers
00330 *
00331                   CALL ZSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ),
00332      $                        1 )
00333 *
00334 *                 Update trailing submatrix within the band and within
00335 *                 the current block. JM is the index of the last column
00336 *                 which needs to be updated.
00337 *
00338                   JM = MIN( JU, J+JB-1 )
00339                   IF( JM.GT.JJ )
00340      $               CALL ZGERU( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
00341      $                           AB( KV, JJ+1 ), LDAB-1,
00342      $                           AB( KV+1, JJ+1 ), LDAB-1 )
00343                ELSE
00344 *
00345 *                 If pivot is zero, set INFO to the index of the pivot
00346 *                 unless a zero pivot has already been found.
00347 *
00348                   IF( INFO.EQ.0 )
00349      $               INFO = JJ
00350                END IF
00351 *
00352 *              Copy current column of A31 into the work array WORK31
00353 *
00354                NW = MIN( JJ-J+1, I3 )
00355                IF( NW.GT.0 )
00356      $            CALL ZCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
00357      $                        WORK31( 1, JJ-J+1 ), 1 )
00358    80       CONTINUE
00359             IF( J+JB.LE.N ) THEN
00360 *
00361 *              Apply the row interchanges to the other blocks.
00362 *
00363                J2 = MIN( JU-J+1, KV ) - JB
00364                J3 = MAX( 0, JU-J-KV+1 )
00365 *
00366 *              Use ZLASWP to apply the row interchanges to A12, A22, and
00367 *              A32.
00368 *
00369                CALL ZLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
00370      $                      IPIV( J ), 1 )
00371 *
00372 *              Adjust the pivot indices.
00373 *
00374                DO 90 I = J, J + JB - 1
00375                   IPIV( I ) = IPIV( I ) + J - 1
00376    90          CONTINUE
00377 *
00378 *              Apply the row interchanges to A13, A23, and A33
00379 *              columnwise.
00380 *
00381                K2 = J - 1 + JB + J2
00382                DO 110 I = 1, J3
00383                   JJ = K2 + I
00384                   DO 100 II = J + I - 1, J + JB - 1
00385                      IP = IPIV( II )
00386                      IF( IP.NE.II ) THEN
00387                         TEMP = AB( KV+1+II-JJ, JJ )
00388                         AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
00389                         AB( KV+1+IP-JJ, JJ ) = TEMP
00390                      END IF
00391   100             CONTINUE
00392   110          CONTINUE
00393 *
00394 *              Update the relevant part of the trailing submatrix
00395 *
00396                IF( J2.GT.0 ) THEN
00397 *
00398 *                 Update A12
00399 *
00400                   CALL ZTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
00401      $                        JB, J2, ONE, AB( KV+1, J ), LDAB-1,
00402      $                        AB( KV+1-JB, J+JB ), LDAB-1 )
00403 *
00404                   IF( I2.GT.0 ) THEN
00405 *
00406 *                    Update A22
00407 *
00408                      CALL ZGEMM( 'No transpose', 'No transpose', I2, J2,
00409      $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
00410      $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
00411      $                           AB( KV+1, J+JB ), LDAB-1 )
00412                   END IF
00413 *
00414                   IF( I3.GT.0 ) THEN
00415 *
00416 *                    Update A32
00417 *
00418                      CALL ZGEMM( 'No transpose', 'No transpose', I3, J2,
00419      $                           JB, -ONE, WORK31, LDWORK,
00420      $                           AB( KV+1-JB, J+JB ), LDAB-1, ONE,
00421      $                           AB( KV+KL+1-JB, J+JB ), LDAB-1 )
00422                   END IF
00423                END IF
00424 *
00425                IF( J3.GT.0 ) THEN
00426 *
00427 *                 Copy the lower triangle of A13 into the work array
00428 *                 WORK13
00429 *
00430                   DO 130 JJ = 1, J3
00431                      DO 120 II = JJ, JB
00432                         WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
00433   120                CONTINUE
00434   130             CONTINUE
00435 *
00436 *                 Update A13 in the work array
00437 *
00438                   CALL ZTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
00439      $                        JB, J3, ONE, AB( KV+1, J ), LDAB-1,
00440      $                        WORK13, LDWORK )
00441 *
00442                   IF( I2.GT.0 ) THEN
00443 *
00444 *                    Update A23
00445 *
00446                      CALL ZGEMM( 'No transpose', 'No transpose', I2, J3,
00447      $                           JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
00448      $                           WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
00449      $                           LDAB-1 )
00450                   END IF
00451 *
00452                   IF( I3.GT.0 ) THEN
00453 *
00454 *                    Update A33
00455 *
00456                      CALL ZGEMM( 'No transpose', 'No transpose', I3, J3,
00457      $                           JB, -ONE, WORK31, LDWORK, WORK13,
00458      $                           LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
00459                   END IF
00460 *
00461 *                 Copy the lower triangle of A13 back into place
00462 *
00463                   DO 150 JJ = 1, J3
00464                      DO 140 II = JJ, JB
00465                         AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
00466   140                CONTINUE
00467   150             CONTINUE
00468                END IF
00469             ELSE
00470 *
00471 *              Adjust the pivot indices.
00472 *
00473                DO 160 I = J, J + JB - 1
00474                   IPIV( I ) = IPIV( I ) + J - 1
00475   160          CONTINUE
00476             END IF
00477 *
00478 *           Partially undo the interchanges in the current block to
00479 *           restore the upper triangular form of A31 and copy the upper
00480 *           triangle of A31 back into place
00481 *
00482             DO 170 JJ = J + JB - 1, J, -1
00483                JP = IPIV( JJ ) - JJ + 1
00484                IF( JP.NE.1 ) THEN
00485 *
00486 *                 Apply interchange to columns J to JJ-1
00487 *
00488                   IF( JP+JJ-1.LT.J+KL ) THEN
00489 *
00490 *                    The interchange does not affect A31
00491 *
00492                      CALL ZSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
00493      $                           AB( KV+JP+JJ-J, J ), LDAB-1 )
00494                   ELSE
00495 *
00496 *                    The interchange does affect A31
00497 *
00498                      CALL ZSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
00499      $                           WORK31( JP+JJ-J-KL, 1 ), LDWORK )
00500                   END IF
00501                END IF
00502 *
00503 *              Copy the current column of A31 back into place
00504 *
00505                NW = MIN( I3, JJ-J+1 )
00506                IF( NW.GT.0 )
00507      $            CALL ZCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
00508      $                        AB( KV+KL+1-JJ+J, JJ ), 1 )
00509   170       CONTINUE
00510   180    CONTINUE
00511       END IF
00512 *
00513       RETURN
00514 *
00515 *     End of ZGBTRF
00516 *
00517       END
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