LAPACK: SRC/zgbbrd.f Source File

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00001       SUBROUTINE ZGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
00002      $                   LDQ, PT, LDPT, C, LDC, WORK, RWORK, INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       CHARACTER          VECT
00011       INTEGER            INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
00012 *     ..
00013 *     .. Array Arguments ..
00014       DOUBLE PRECISION   D( * ), E( * ), RWORK( * )
00015       COMPLEX*16         AB( LDAB, * ), C( LDC, * ), PT( LDPT, * ),
00016      $                   Q( LDQ, * ), WORK( * )
00017 *     ..
00018 *
00019 *  Purpose
00020 *  =======
00021 *
00022 *  ZGBBRD reduces a complex general m-by-n band matrix A to real upper
00023 *  bidiagonal form B by a unitary transformation: Q' * A * P = B.
00024 *
00025 *  The routine computes B, and optionally forms Q or P', or computes
00026 *  Q'*C for a given matrix C.
00027 *
00028 *  Arguments
00029 *  =========
00030 *
00031 *  VECT    (input) CHARACTER*1
00032 *          Specifies whether or not the matrices Q and P' are to be
00033 *          formed.
00034 *          = 'N': do not form Q or P';
00035 *          = 'Q': form Q only;
00036 *          = 'P': form P' only;
00037 *          = 'B': form both.
00038 *
00039 *  M       (input) INTEGER
00040 *          The number of rows of the matrix A.  M >= 0.
00041 *
00042 *  N       (input) INTEGER
00043 *          The number of columns of the matrix A.  N >= 0.
00044 *
00045 *  NCC     (input) INTEGER
00046 *          The number of columns of the matrix C.  NCC >= 0.
00047 *
00048 *  KL      (input) INTEGER
00049 *          The number of subdiagonals of the matrix A. KL >= 0.
00050 *
00051 *  KU      (input) INTEGER
00052 *          The number of superdiagonals of the matrix A. KU >= 0.
00053 *
00054 *  AB      (input/output) COMPLEX*16 array, dimension (LDAB,N)
00055 *          On entry, the m-by-n band matrix A, stored in rows 1 to
00056 *          KL+KU+1. The j-th column of A is stored in the j-th column of
00057 *          the array AB as follows:
00058 *          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
00059 *          On exit, A is overwritten by values generated during the
00060 *          reduction.
00061 *
00062 *  LDAB    (input) INTEGER
00063 *          The leading dimension of the array A. LDAB >= KL+KU+1.
00064 *
00065 *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
00066 *          The diagonal elements of the bidiagonal matrix B.
00067 *
00068 *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
00069 *          The superdiagonal elements of the bidiagonal matrix B.
00070 *
00071 *  Q       (output) COMPLEX*16 array, dimension (LDQ,M)
00072 *          If VECT = 'Q' or 'B', the m-by-m unitary matrix Q.
00073 *          If VECT = 'N' or 'P', the array Q is not referenced.
00074 *
00075 *  LDQ     (input) INTEGER
00076 *          The leading dimension of the array Q.
00077 *          LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
00078 *
00079 *  PT      (output) COMPLEX*16 array, dimension (LDPT,N)
00080 *          If VECT = 'P' or 'B', the n-by-n unitary matrix P'.
00081 *          If VECT = 'N' or 'Q', the array PT is not referenced.
00082 *
00083 *  LDPT    (input) INTEGER
00084 *          The leading dimension of the array PT.
00085 *          LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
00086 *
00087 *  C       (input/output) COMPLEX*16 array, dimension (LDC,NCC)
00088 *          On entry, an m-by-ncc matrix C.
00089 *          On exit, C is overwritten by Q'*C.
00090 *          C is not referenced if NCC = 0.
00091 *
00092 *  LDC     (input) INTEGER
00093 *          The leading dimension of the array C.
00094 *          LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
00095 *
00096 *  WORK    (workspace) COMPLEX*16 array, dimension (max(M,N))
00097 *
00098 *  RWORK   (workspace) DOUBLE PRECISION array, dimension (max(M,N))
00099 *
00100 *  INFO    (output) INTEGER
00101 *          = 0:  successful exit.
00102 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00103 *
00104 *  =====================================================================
00105 *
00106 *     .. Parameters ..
00107       DOUBLE PRECISION   ZERO
00108       PARAMETER          ( ZERO = 0.0D+0 )
00109       COMPLEX*16         CZERO, CONE
00110       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00111      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00112 *     ..
00113 *     .. Local Scalars ..
00114       LOGICAL            WANTB, WANTC, WANTPT, WANTQ
00115       INTEGER            I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
00116      $                   KUN, L, MINMN, ML, ML0, MU, MU0, NR, NRT
00117       DOUBLE PRECISION   ABST, RC
00118       COMPLEX*16         RA, RB, RS, T
00119 *     ..
00120 *     .. External Subroutines ..
00121       EXTERNAL           XERBLA, ZLARGV, ZLARTG, ZLARTV, ZLASET, ZROT,
00122      $                   ZSCAL
00123 *     ..
00124 *     .. Intrinsic Functions ..
00125       INTRINSIC          ABS, DCONJG, MAX, MIN
00126 *     ..
00127 *     .. External Functions ..
00128       LOGICAL            LSAME
00129       EXTERNAL           LSAME
00130 *     ..
00131 *     .. Executable Statements ..
00132 *
00133 *     Test the input parameters
00134 *
00135       WANTB = LSAME( VECT, 'B' )
00136       WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
00137       WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
00138       WANTC = NCC.GT.0
00139       KLU1 = KL + KU + 1
00140       INFO = 0
00141       IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
00142      $     THEN
00143          INFO = -1
00144       ELSE IF( M.LT.0 ) THEN
00145          INFO = -2
00146       ELSE IF( N.LT.0 ) THEN
00147          INFO = -3
00148       ELSE IF( NCC.LT.0 ) THEN
00149          INFO = -4
00150       ELSE IF( KL.LT.0 ) THEN
00151          INFO = -5
00152       ELSE IF( KU.LT.0 ) THEN
00153          INFO = -6
00154       ELSE IF( LDAB.LT.KLU1 ) THEN
00155          INFO = -8
00156       ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
00157          INFO = -12
00158       ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
00159          INFO = -14
00160       ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
00161          INFO = -16
00162       END IF
00163       IF( INFO.NE.0 ) THEN
00164          CALL XERBLA( 'ZGBBRD', -INFO )
00165          RETURN
00166       END IF
00167 *
00168 *     Initialize Q and P' to the unit matrix, if needed
00169 *
00170       IF( WANTQ )
00171      $   CALL ZLASET( 'Full', M, M, CZERO, CONE, Q, LDQ )
00172       IF( WANTPT )
00173      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, PT, LDPT )
00174 *
00175 *     Quick return if possible.
00176 *
00177       IF( M.EQ.0 .OR. N.EQ.0 )
00178      $   RETURN
00179 *
00180       MINMN = MIN( M, N )
00181 *
00182       IF( KL+KU.GT.1 ) THEN
00183 *
00184 *        Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
00185 *        first to lower bidiagonal form and then transform to upper
00186 *        bidiagonal
00187 *
00188          IF( KU.GT.0 ) THEN
00189             ML0 = 1
00190             MU0 = 2
00191          ELSE
00192             ML0 = 2
00193             MU0 = 1
00194          END IF
00195 *
00196 *        Wherever possible, plane rotations are generated and applied in
00197 *        vector operations of length NR over the index set J1:J2:KLU1.
00198 *
00199 *        The complex sines of the plane rotations are stored in WORK,
00200 *        and the real cosines in RWORK.
00201 *
00202          KLM = MIN( M-1, KL )
00203          KUN = MIN( N-1, KU )
00204          KB = KLM + KUN
00205          KB1 = KB + 1
00206          INCA = KB1*LDAB
00207          NR = 0
00208          J1 = KLM + 2
00209          J2 = 1 - KUN
00210 *
00211          DO 90 I = 1, MINMN
00212 *
00213 *           Reduce i-th column and i-th row of matrix to bidiagonal form
00214 *
00215             ML = KLM + 1
00216             MU = KUN + 1
00217             DO 80 KK = 1, KB
00218                J1 = J1 + KB
00219                J2 = J2 + KB
00220 *
00221 *              generate plane rotations to annihilate nonzero elements
00222 *              which have been created below the band
00223 *
00224                IF( NR.GT.0 )
00225      $            CALL ZLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
00226      $                         WORK( J1 ), KB1, RWORK( J1 ), KB1 )
00227 *
00228 *              apply plane rotations from the left
00229 *
00230                DO 10 L = 1, KB
00231                   IF( J2-KLM+L-1.GT.N ) THEN
00232                      NRT = NR - 1
00233                   ELSE
00234                      NRT = NR
00235                   END IF
00236                   IF( NRT.GT.0 )
00237      $               CALL ZLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
00238      $                            AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
00239      $                            RWORK( J1 ), WORK( J1 ), KB1 )
00240    10          CONTINUE
00241 *
00242                IF( ML.GT.ML0 ) THEN
00243                   IF( ML.LE.M-I+1 ) THEN
00244 *
00245 *                    generate plane rotation to annihilate a(i+ml-1,i)
00246 *                    within the band, and apply rotation from the left
00247 *
00248                      CALL ZLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
00249      $                            RWORK( I+ML-1 ), WORK( I+ML-1 ), RA )
00250                      AB( KU+ML-1, I ) = RA
00251                      IF( I.LT.N )
00252      $                  CALL ZROT( MIN( KU+ML-2, N-I ),
00253      $                             AB( KU+ML-2, I+1 ), LDAB-1,
00254      $                             AB( KU+ML-1, I+1 ), LDAB-1,
00255      $                             RWORK( I+ML-1 ), WORK( I+ML-1 ) )
00256                   END IF
00257                   NR = NR + 1
00258                   J1 = J1 - KB1
00259                END IF
00260 *
00261                IF( WANTQ ) THEN
00262 *
00263 *                 accumulate product of plane rotations in Q
00264 *
00265                   DO 20 J = J1, J2, KB1
00266                      CALL ZROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
00267      $                          RWORK( J ), DCONJG( WORK( J ) ) )
00268    20             CONTINUE
00269                END IF
00270 *
00271                IF( WANTC ) THEN
00272 *
00273 *                 apply plane rotations to C
00274 *
00275                   DO 30 J = J1, J2, KB1
00276                      CALL ZROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
00277      $                          RWORK( J ), WORK( J ) )
00278    30             CONTINUE
00279                END IF
00280 *
00281                IF( J2+KUN.GT.N ) THEN
00282 *
00283 *                 adjust J2 to keep within the bounds of the matrix
00284 *
00285                   NR = NR - 1
00286                   J2 = J2 - KB1
00287                END IF
00288 *
00289                DO 40 J = J1, J2, KB1
00290 *
00291 *                 create nonzero element a(j-1,j+ku) above the band
00292 *                 and store it in WORK(n+1:2*n)
00293 *
00294                   WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
00295                   AB( 1, J+KUN ) = RWORK( J )*AB( 1, J+KUN )
00296    40          CONTINUE
00297 *
00298 *              generate plane rotations to annihilate nonzero elements
00299 *              which have been generated above the band
00300 *
00301                IF( NR.GT.0 )
00302      $            CALL ZLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
00303      $                         WORK( J1+KUN ), KB1, RWORK( J1+KUN ),
00304      $                         KB1 )
00305 *
00306 *              apply plane rotations from the right
00307 *
00308                DO 50 L = 1, KB
00309                   IF( J2+L-1.GT.M ) THEN
00310                      NRT = NR - 1
00311                   ELSE
00312                      NRT = NR
00313                   END IF
00314                   IF( NRT.GT.0 )
00315      $               CALL ZLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
00316      $                            AB( L, J1+KUN ), INCA,
00317      $                            RWORK( J1+KUN ), WORK( J1+KUN ), KB1 )
00318    50          CONTINUE
00319 *
00320                IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
00321                   IF( MU.LE.N-I+1 ) THEN
00322 *
00323 *                    generate plane rotation to annihilate a(i,i+mu-1)
00324 *                    within the band, and apply rotation from the right
00325 *
00326                      CALL ZLARTG( AB( KU-MU+3, I+MU-2 ),
00327      $                            AB( KU-MU+2, I+MU-1 ),
00328      $                            RWORK( I+MU-1 ), WORK( I+MU-1 ), RA )
00329                      AB( KU-MU+3, I+MU-2 ) = RA
00330                      CALL ZROT( MIN( KL+MU-2, M-I ),
00331      $                          AB( KU-MU+4, I+MU-2 ), 1,
00332      $                          AB( KU-MU+3, I+MU-1 ), 1,
00333      $                          RWORK( I+MU-1 ), WORK( I+MU-1 ) )
00334                   END IF
00335                   NR = NR + 1
00336                   J1 = J1 - KB1
00337                END IF
00338 *
00339                IF( WANTPT ) THEN
00340 *
00341 *                 accumulate product of plane rotations in P'
00342 *
00343                   DO 60 J = J1, J2, KB1
00344                      CALL ZROT( N, PT( J+KUN-1, 1 ), LDPT,
00345      $                          PT( J+KUN, 1 ), LDPT, RWORK( J+KUN ),
00346      $                          DCONJG( WORK( J+KUN ) ) )
00347    60             CONTINUE
00348                END IF
00349 *
00350                IF( J2+KB.GT.M ) THEN
00351 *
00352 *                 adjust J2 to keep within the bounds of the matrix
00353 *
00354                   NR = NR - 1
00355                   J2 = J2 - KB1
00356                END IF
00357 *
00358                DO 70 J = J1, J2, KB1
00359 *
00360 *                 create nonzero element a(j+kl+ku,j+ku-1) below the
00361 *                 band and store it in WORK(1:n)
00362 *
00363                   WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
00364                   AB( KLU1, J+KUN ) = RWORK( J+KUN )*AB( KLU1, J+KUN )
00365    70          CONTINUE
00366 *
00367                IF( ML.GT.ML0 ) THEN
00368                   ML = ML - 1
00369                ELSE
00370                   MU = MU - 1
00371                END IF
00372    80       CONTINUE
00373    90    CONTINUE
00374       END IF
00375 *
00376       IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
00377 *
00378 *        A has been reduced to complex lower bidiagonal form
00379 *
00380 *        Transform lower bidiagonal form to upper bidiagonal by applying
00381 *        plane rotations from the left, overwriting superdiagonal
00382 *        elements on subdiagonal elements
00383 *
00384          DO 100 I = 1, MIN( M-1, N )
00385             CALL ZLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
00386             AB( 1, I ) = RA
00387             IF( I.LT.N ) THEN
00388                AB( 2, I ) = RS*AB( 1, I+1 )
00389                AB( 1, I+1 ) = RC*AB( 1, I+1 )
00390             END IF
00391             IF( WANTQ )
00392      $         CALL ZROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC,
00393      $                    DCONJG( RS ) )
00394             IF( WANTC )
00395      $         CALL ZROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
00396      $                    RS )
00397   100    CONTINUE
00398       ELSE
00399 *
00400 *        A has been reduced to complex upper bidiagonal form or is
00401 *        diagonal
00402 *
00403          IF( KU.GT.0 .AND. M.LT.N ) THEN
00404 *
00405 *           Annihilate a(m,m+1) by applying plane rotations from the
00406 *           right
00407 *
00408             RB = AB( KU, M+1 )
00409             DO 110 I = M, 1, -1
00410                CALL ZLARTG( AB( KU+1, I ), RB, RC, RS, RA )
00411                AB( KU+1, I ) = RA
00412                IF( I.GT.1 ) THEN
00413                   RB = -DCONJG( RS )*AB( KU, I )
00414                   AB( KU, I ) = RC*AB( KU, I )
00415                END IF
00416                IF( WANTPT )
00417      $            CALL ZROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
00418      $                       RC, DCONJG( RS ) )
00419   110       CONTINUE
00420          END IF
00421       END IF
00422 *
00423 *     Make diagonal and superdiagonal elements real, storing them in D
00424 *     and E
00425 *
00426       T = AB( KU+1, 1 )
00427       DO 120 I = 1, MINMN
00428          ABST = ABS( T )
00429          D( I ) = ABST
00430          IF( ABST.NE.ZERO ) THEN
00431             T = T / ABST
00432          ELSE
00433             T = CONE
00434          END IF
00435          IF( WANTQ )
00436      $      CALL ZSCAL( M, T, Q( 1, I ), 1 )
00437          IF( WANTC )
00438      $      CALL ZSCAL( NCC, DCONJG( T ), C( I, 1 ), LDC )
00439          IF( I.LT.MINMN ) THEN
00440             IF( KU.EQ.0 .AND. KL.EQ.0 ) THEN
00441                E( I ) = ZERO
00442                T = AB( 1, I+1 )
00443             ELSE
00444                IF( KU.EQ.0 ) THEN
00445                   T = AB( 2, I )*DCONJG( T )
00446                ELSE
00447                   T = AB( KU, I+1 )*DCONJG( T )
00448                END IF
00449                ABST = ABS( T )
00450                E( I ) = ABST
00451                IF( ABST.NE.ZERO ) THEN
00452                   T = T / ABST
00453                ELSE
00454                   T = CONE
00455                END IF
00456                IF( WANTPT )
00457      $            CALL ZSCAL( N, T, PT( I+1, 1 ), LDPT )
00458                T = AB( KU+1, I+1 )*DCONJG( T )
00459             END IF
00460          END IF
00461   120 CONTINUE
00462       RETURN
00463 *
00464 *     End of ZGBBRD
00465 *
00466       END