LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cgbbrd.f
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00001 *> \brief \b CGBBRD
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CGBBRD + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgbbrd.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgbbrd.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
00022 *                          LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       CHARACTER          VECT
00026 *       INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       REAL               D( * ), E( * ), RWORK( * )
00030 *       COMPLEX            AB( LDAB, * ), C( LDC, * ), PT( LDPT, * ),
00031 *      $                   Q( LDQ, * ), WORK( * )
00032 *       ..
00033 *  
00034 *
00035 *> \par Purpose:
00036 *  =============
00037 *>
00038 *> \verbatim
00039 *>
00040 *> CGBBRD reduces a complex general m-by-n band matrix A to real upper
00041 *> bidiagonal form B by a unitary transformation: Q**H * A * P = B.
00042 *>
00043 *> The routine computes B, and optionally forms Q or P**H, or computes
00044 *> Q**H*C for a given matrix C.
00045 *> \endverbatim
00046 *
00047 *  Arguments:
00048 *  ==========
00049 *
00050 *> \param[in] VECT
00051 *> \verbatim
00052 *>          VECT is CHARACTER*1
00053 *>          Specifies whether or not the matrices Q and P**H are to be
00054 *>          formed.
00055 *>          = 'N': do not form Q or P**H;
00056 *>          = 'Q': form Q only;
00057 *>          = 'P': form P**H only;
00058 *>          = 'B': form both.
00059 *> \endverbatim
00060 *>
00061 *> \param[in] M
00062 *> \verbatim
00063 *>          M is INTEGER
00064 *>          The number of rows of the matrix A.  M >= 0.
00065 *> \endverbatim
00066 *>
00067 *> \param[in] N
00068 *> \verbatim
00069 *>          N is INTEGER
00070 *>          The number of columns of the matrix A.  N >= 0.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] NCC
00074 *> \verbatim
00075 *>          NCC is INTEGER
00076 *>          The number of columns of the matrix C.  NCC >= 0.
00077 *> \endverbatim
00078 *>
00079 *> \param[in] KL
00080 *> \verbatim
00081 *>          KL is INTEGER
00082 *>          The number of subdiagonals of the matrix A. KL >= 0.
00083 *> \endverbatim
00084 *>
00085 *> \param[in] KU
00086 *> \verbatim
00087 *>          KU is INTEGER
00088 *>          The number of superdiagonals of the matrix A. KU >= 0.
00089 *> \endverbatim
00090 *>
00091 *> \param[in,out] AB
00092 *> \verbatim
00093 *>          AB is COMPLEX array, dimension (LDAB,N)
00094 *>          On entry, the m-by-n band matrix A, stored in rows 1 to
00095 *>          KL+KU+1. The j-th column of A is stored in the j-th column of
00096 *>          the array AB as follows:
00097 *>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
00098 *>          On exit, A is overwritten by values generated during the
00099 *>          reduction.
00100 *> \endverbatim
00101 *>
00102 *> \param[in] LDAB
00103 *> \verbatim
00104 *>          LDAB is INTEGER
00105 *>          The leading dimension of the array A. LDAB >= KL+KU+1.
00106 *> \endverbatim
00107 *>
00108 *> \param[out] D
00109 *> \verbatim
00110 *>          D is REAL array, dimension (min(M,N))
00111 *>          The diagonal elements of the bidiagonal matrix B.
00112 *> \endverbatim
00113 *>
00114 *> \param[out] E
00115 *> \verbatim
00116 *>          E is REAL array, dimension (min(M,N)-1)
00117 *>          The superdiagonal elements of the bidiagonal matrix B.
00118 *> \endverbatim
00119 *>
00120 *> \param[out] Q
00121 *> \verbatim
00122 *>          Q is COMPLEX array, dimension (LDQ,M)
00123 *>          If VECT = 'Q' or 'B', the m-by-m unitary matrix Q.
00124 *>          If VECT = 'N' or 'P', the array Q is not referenced.
00125 *> \endverbatim
00126 *>
00127 *> \param[in] LDQ
00128 *> \verbatim
00129 *>          LDQ is INTEGER
00130 *>          The leading dimension of the array Q.
00131 *>          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
00132 *> \endverbatim
00133 *>
00134 *> \param[out] PT
00135 *> \verbatim
00136 *>          PT is COMPLEX array, dimension (LDPT,N)
00137 *>          If VECT = 'P' or 'B', the n-by-n unitary matrix P'.
00138 *>          If VECT = 'N' or 'Q', the array PT is not referenced.
00139 *> \endverbatim
00140 *>
00141 *> \param[in] LDPT
00142 *> \verbatim
00143 *>          LDPT is INTEGER
00144 *>          The leading dimension of the array PT.
00145 *>          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
00146 *> \endverbatim
00147 *>
00148 *> \param[in,out] C
00149 *> \verbatim
00150 *>          C is COMPLEX array, dimension (LDC,NCC)
00151 *>          On entry, an m-by-ncc matrix C.
00152 *>          On exit, C is overwritten by Q**H*C.
00153 *>          C is not referenced if NCC = 0.
00154 *> \endverbatim
00155 *>
00156 *> \param[in] LDC
00157 *> \verbatim
00158 *>          LDC is INTEGER
00159 *>          The leading dimension of the array C.
00160 *>          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
00161 *> \endverbatim
00162 *>
00163 *> \param[out] WORK
00164 *> \verbatim
00165 *>          WORK is COMPLEX array, dimension (max(M,N))
00166 *> \endverbatim
00167 *>
00168 *> \param[out] RWORK
00169 *> \verbatim
00170 *>          RWORK is REAL array, dimension (max(M,N))
00171 *> \endverbatim
00172 *>
00173 *> \param[out] INFO
00174 *> \verbatim
00175 *>          INFO is INTEGER
00176 *>          = 0:  successful exit.
00177 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00178 *> \endverbatim
00179 *
00180 *  Authors:
00181 *  ========
00182 *
00183 *> \author Univ. of Tennessee 
00184 *> \author Univ. of California Berkeley 
00185 *> \author Univ. of Colorado Denver 
00186 *> \author NAG Ltd. 
00187 *
00188 *> \date November 2011
00189 *
00190 *> \ingroup complexGBcomputational
00191 *
00192 *  =====================================================================
00193       SUBROUTINE CGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
00194      $                   LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )
00195 *
00196 *  -- LAPACK computational routine (version 3.4.0) --
00197 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00198 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00199 *     November 2011
00200 *
00201 *     .. Scalar Arguments ..
00202       CHARACTER          VECT
00203       INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
00204 *     ..
00205 *     .. Array Arguments ..
00206       REAL               D( * ), E( * ), RWORK( * )
00207       COMPLEX            AB( LDAB, * ), C( LDC, * ), PT( LDPT, * ),
00208      $                   Q( LDQ, * ), WORK( * )
00209 *     ..
00210 *
00211 *  =====================================================================
00212 *
00213 *     .. Parameters ..
00214       REAL               ZERO
00215       PARAMETER          ( ZERO = 0.0E+0 )
00216       COMPLEX            CZERO, CONE
00217       PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ),
00218      $                   CONE = ( 1.0E+0, 0.0E+0 ) )
00219 *     ..
00220 *     .. Local Scalars ..
00221       LOGICAL            WANTB, WANTC, WANTPT, WANTQ
00222       INTEGER            I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
00223      $                   KUN, L, MINMN, ML, ML0, MU, MU0, NR, NRT
00224       REAL               ABST, RC
00225       COMPLEX            RA, RB, RS, T
00226 *     ..
00227 *     .. External Subroutines ..
00228       EXTERNAL           CLARGV, CLARTG, CLARTV, CLASET, CROT, CSCAL,
00229      $                   XERBLA
00230 *     ..
00231 *     .. Intrinsic Functions ..
00232       INTRINSIC          ABS, CONJG, MAX, MIN
00233 *     ..
00234 *     .. External Functions ..
00235       LOGICAL            LSAME
00236       EXTERNAL           LSAME
00237 *     ..
00238 *     .. Executable Statements ..
00239 *
00240 *     Test the input parameters
00241 *
00242       WANTB = LSAME( VECT, 'B' )
00243       WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
00244       WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
00245       WANTC = NCC.GT.0
00246       KLU1 = KL + KU + 1
00247       INFO = 0
00248       IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
00249      $     THEN
00250          INFO = -1
00251       ELSE IF( M.LT.0 ) THEN
00252          INFO = -2
00253       ELSE IF( N.LT.0 ) THEN
00254          INFO = -3
00255       ELSE IF( NCC.LT.0 ) THEN
00256          INFO = -4
00257       ELSE IF( KL.LT.0 ) THEN
00258          INFO = -5
00259       ELSE IF( KU.LT.0 ) THEN
00260          INFO = -6
00261       ELSE IF( LDAB.LT.KLU1 ) THEN
00262          INFO = -8
00263       ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
00264          INFO = -12
00265       ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
00266          INFO = -14
00267       ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
00268          INFO = -16
00269       END IF
00270       IF( INFO.NE.0 ) THEN
00271          CALL XERBLA( 'CGBBRD', -INFO )
00272          RETURN
00273       END IF
00274 *
00275 *     Initialize Q and P**H to the unit matrix, if needed
00276 *
00277       IF( WANTQ )
00278      $   CALL CLASET( 'Full', M, M, CZERO, CONE, Q, LDQ )
00279       IF( WANTPT )
00280      $   CALL CLASET( 'Full', N, N, CZERO, CONE, PT, LDPT )
00281 *
00282 *     Quick return if possible.
00283 *
00284       IF( M.EQ.0 .OR. N.EQ.0 )
00285      $   RETURN
00286 *
00287       MINMN = MIN( M, N )
00288 *
00289       IF( KL+KU.GT.1 ) THEN
00290 *
00291 *        Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
00292 *        first to lower bidiagonal form and then transform to upper
00293 *        bidiagonal
00294 *
00295          IF( KU.GT.0 ) THEN
00296             ML0 = 1
00297             MU0 = 2
00298          ELSE
00299             ML0 = 2
00300             MU0 = 1
00301          END IF
00302 *
00303 *        Wherever possible, plane rotations are generated and applied in
00304 *        vector operations of length NR over the index set J1:J2:KLU1.
00305 *
00306 *        The complex sines of the plane rotations are stored in WORK,
00307 *        and the real cosines in RWORK.
00308 *
00309          KLM = MIN( M-1, KL )
00310          KUN = MIN( N-1, KU )
00311          KB = KLM + KUN
00312          KB1 = KB + 1
00313          INCA = KB1*LDAB
00314          NR = 0
00315          J1 = KLM + 2
00316          J2 = 1 - KUN
00317 *
00318          DO 90 I = 1, MINMN
00319 *
00320 *           Reduce i-th column and i-th row of matrix to bidiagonal form
00321 *
00322             ML = KLM + 1
00323             MU = KUN + 1
00324             DO 80 KK = 1, KB
00325                J1 = J1 + KB
00326                J2 = J2 + KB
00327 *
00328 *              generate plane rotations to annihilate nonzero elements
00329 *              which have been created below the band
00330 *
00331                IF( NR.GT.0 )
00332      $            CALL CLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
00333      $                         WORK( J1 ), KB1, RWORK( J1 ), KB1 )
00334 *
00335 *              apply plane rotations from the left
00336 *
00337                DO 10 L = 1, KB
00338                   IF( J2-KLM+L-1.GT.N ) THEN
00339                      NRT = NR - 1
00340                   ELSE
00341                      NRT = NR
00342                   END IF
00343                   IF( NRT.GT.0 )
00344      $               CALL CLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
00345      $                            AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
00346      $                            RWORK( J1 ), WORK( J1 ), KB1 )
00347    10          CONTINUE
00348 *
00349                IF( ML.GT.ML0 ) THEN
00350                   IF( ML.LE.M-I+1 ) THEN
00351 *
00352 *                    generate plane rotation to annihilate a(i+ml-1,i)
00353 *                    within the band, and apply rotation from the left
00354 *
00355                      CALL CLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
00356      $                            RWORK( I+ML-1 ), WORK( I+ML-1 ), RA )
00357                      AB( KU+ML-1, I ) = RA
00358                      IF( I.LT.N )
00359      $                  CALL CROT( MIN( KU+ML-2, N-I ),
00360      $                             AB( KU+ML-2, I+1 ), LDAB-1,
00361      $                             AB( KU+ML-1, I+1 ), LDAB-1,
00362      $                             RWORK( I+ML-1 ), WORK( I+ML-1 ) )
00363                   END IF
00364                   NR = NR + 1
00365                   J1 = J1 - KB1
00366                END IF
00367 *
00368                IF( WANTQ ) THEN
00369 *
00370 *                 accumulate product of plane rotations in Q
00371 *
00372                   DO 20 J = J1, J2, KB1
00373                      CALL CROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
00374      $                          RWORK( J ), CONJG( WORK( J ) ) )
00375    20             CONTINUE
00376                END IF
00377 *
00378                IF( WANTC ) THEN
00379 *
00380 *                 apply plane rotations to C
00381 *
00382                   DO 30 J = J1, J2, KB1
00383                      CALL CROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
00384      $                          RWORK( J ), WORK( J ) )
00385    30             CONTINUE
00386                END IF
00387 *
00388                IF( J2+KUN.GT.N ) THEN
00389 *
00390 *                 adjust J2 to keep within the bounds of the matrix
00391 *
00392                   NR = NR - 1
00393                   J2 = J2 - KB1
00394                END IF
00395 *
00396                DO 40 J = J1, J2, KB1
00397 *
00398 *                 create nonzero element a(j-1,j+ku) above the band
00399 *                 and store it in WORK(n+1:2*n)
00400 *
00401                   WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
00402                   AB( 1, J+KUN ) = RWORK( J )*AB( 1, J+KUN )
00403    40          CONTINUE
00404 *
00405 *              generate plane rotations to annihilate nonzero elements
00406 *              which have been generated above the band
00407 *
00408                IF( NR.GT.0 )
00409      $            CALL CLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
00410      $                         WORK( J1+KUN ), KB1, RWORK( J1+KUN ),
00411      $                         KB1 )
00412 *
00413 *              apply plane rotations from the right
00414 *
00415                DO 50 L = 1, KB
00416                   IF( J2+L-1.GT.M ) THEN
00417                      NRT = NR - 1
00418                   ELSE
00419                      NRT = NR
00420                   END IF
00421                   IF( NRT.GT.0 )
00422      $               CALL CLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
00423      $                            AB( L, J1+KUN ), INCA,
00424      $                            RWORK( J1+KUN ), WORK( J1+KUN ), KB1 )
00425    50          CONTINUE
00426 *
00427                IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
00428                   IF( MU.LE.N-I+1 ) THEN
00429 *
00430 *                    generate plane rotation to annihilate a(i,i+mu-1)
00431 *                    within the band, and apply rotation from the right
00432 *
00433                      CALL CLARTG( AB( KU-MU+3, I+MU-2 ),
00434      $                            AB( KU-MU+2, I+MU-1 ),
00435      $                            RWORK( I+MU-1 ), WORK( I+MU-1 ), RA )
00436                      AB( KU-MU+3, I+MU-2 ) = RA
00437                      CALL CROT( MIN( KL+MU-2, M-I ),
00438      $                          AB( KU-MU+4, I+MU-2 ), 1,
00439      $                          AB( KU-MU+3, I+MU-1 ), 1,
00440      $                          RWORK( I+MU-1 ), WORK( I+MU-1 ) )
00441                   END IF
00442                   NR = NR + 1
00443                   J1 = J1 - KB1
00444                END IF
00445 *
00446                IF( WANTPT ) THEN
00447 *
00448 *                 accumulate product of plane rotations in P**H
00449 *
00450                   DO 60 J = J1, J2, KB1
00451                      CALL CROT( N, PT( J+KUN-1, 1 ), LDPT,
00452      $                          PT( J+KUN, 1 ), LDPT, RWORK( J+KUN ),
00453      $                          CONJG( WORK( J+KUN ) ) )
00454    60             CONTINUE
00455                END IF
00456 *
00457                IF( J2+KB.GT.M ) THEN
00458 *
00459 *                 adjust J2 to keep within the bounds of the matrix
00460 *
00461                   NR = NR - 1
00462                   J2 = J2 - KB1
00463                END IF
00464 *
00465                DO 70 J = J1, J2, KB1
00466 *
00467 *                 create nonzero element a(j+kl+ku,j+ku-1) below the
00468 *                 band and store it in WORK(1:n)
00469 *
00470                   WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
00471                   AB( KLU1, J+KUN ) = RWORK( J+KUN )*AB( KLU1, J+KUN )
00472    70          CONTINUE
00473 *
00474                IF( ML.GT.ML0 ) THEN
00475                   ML = ML - 1
00476                ELSE
00477                   MU = MU - 1
00478                END IF
00479    80       CONTINUE
00480    90    CONTINUE
00481       END IF
00482 *
00483       IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
00484 *
00485 *        A has been reduced to complex lower bidiagonal form
00486 *
00487 *        Transform lower bidiagonal form to upper bidiagonal by applying
00488 *        plane rotations from the left, overwriting superdiagonal
00489 *        elements on subdiagonal elements
00490 *
00491          DO 100 I = 1, MIN( M-1, N )
00492             CALL CLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
00493             AB( 1, I ) = RA
00494             IF( I.LT.N ) THEN
00495                AB( 2, I ) = RS*AB( 1, I+1 )
00496                AB( 1, I+1 ) = RC*AB( 1, I+1 )
00497             END IF
00498             IF( WANTQ )
00499      $         CALL CROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC,
00500      $                    CONJG( RS ) )
00501             IF( WANTC )
00502      $         CALL CROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
00503      $                    RS )
00504   100    CONTINUE
00505       ELSE
00506 *
00507 *        A has been reduced to complex upper bidiagonal form or is
00508 *        diagonal
00509 *
00510          IF( KU.GT.0 .AND. M.LT.N ) THEN
00511 *
00512 *           Annihilate a(m,m+1) by applying plane rotations from the
00513 *           right
00514 *
00515             RB = AB( KU, M+1 )
00516             DO 110 I = M, 1, -1
00517                CALL CLARTG( AB( KU+1, I ), RB, RC, RS, RA )
00518                AB( KU+1, I ) = RA
00519                IF( I.GT.1 ) THEN
00520                   RB = -CONJG( RS )*AB( KU, I )
00521                   AB( KU, I ) = RC*AB( KU, I )
00522                END IF
00523                IF( WANTPT )
00524      $            CALL CROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
00525      $                       RC, CONJG( RS ) )
00526   110       CONTINUE
00527          END IF
00528       END IF
00529 *
00530 *     Make diagonal and superdiagonal elements real, storing them in D
00531 *     and E
00532 *
00533       T = AB( KU+1, 1 )
00534       DO 120 I = 1, MINMN
00535          ABST = ABS( T )
00536          D( I ) = ABST
00537          IF( ABST.NE.ZERO ) THEN
00538             T = T / ABST
00539          ELSE
00540             T = CONE
00541          END IF
00542          IF( WANTQ )
00543      $      CALL CSCAL( M, T, Q( 1, I ), 1 )
00544          IF( WANTC )
00545      $      CALL CSCAL( NCC, CONJG( T ), C( I, 1 ), LDC )
00546          IF( I.LT.MINMN ) THEN
00547             IF( KU.EQ.0 .AND. KL.EQ.0 ) THEN
00548                E( I ) = ZERO
00549                T = AB( 1, I+1 )
00550             ELSE
00551                IF( KU.EQ.0 ) THEN
00552                   T = AB( 2, I )*CONJG( T )
00553                ELSE
00554                   T = AB( KU, I+1 )*CONJG( T )
00555                END IF
00556                ABST = ABS( T )
00557                E( I ) = ABST
00558                IF( ABST.NE.ZERO ) THEN
00559                   T = T / ABST
00560                ELSE
00561                   T = CONE
00562                END IF
00563                IF( WANTPT )
00564      $            CALL CSCAL( N, T, PT( I+1, 1 ), LDPT )
00565                T = AB( KU+1, I+1 )*CONJG( T )
00566             END IF
00567          END IF
00568   120 CONTINUE
00569       RETURN
00570 *
00571 *     End of CGBBRD
00572 *
00573       END
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