LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zpbstf.f
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00001 *> \brief \b ZPBSTF
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, KD, LDAB, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       COMPLEX*16         AB( LDAB, * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> ZPBSTF computes a split Cholesky factorization of a complex
00038 *> Hermitian positive definite band matrix A.
00039 *>
00040 *> This routine is designed to be used in conjunction with ZHBGST.
00041 *>
00042 *> The factorization has the form  A = S**H*S  where S is a band matrix
00043 *> of the same bandwidth as A and the following structure:
00044 *>
00045 *>   S = ( U    )
00046 *>       ( M  L )
00047 *>
00048 *> where U is upper triangular of order m = (n+kd)/2, and L is lower
00049 *> triangular of order n-m.
00050 *> \endverbatim
00051 *
00052 *  Arguments:
00053 *  ==========
00054 *
00055 *> \param[in] UPLO
00056 *> \verbatim
00057 *>          UPLO is CHARACTER*1
00058 *>          = 'U':  Upper triangle of A is stored;
00059 *>          = 'L':  Lower triangle of A is stored.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] N
00063 *> \verbatim
00064 *>          N is INTEGER
00065 *>          The order of the matrix A.  N >= 0.
00066 *> \endverbatim
00067 *>
00068 *> \param[in] KD
00069 *> \verbatim
00070 *>          KD is INTEGER
00071 *>          The number of superdiagonals of the matrix A if UPLO = 'U',
00072 *>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
00073 *> \endverbatim
00074 *>
00075 *> \param[in,out] AB
00076 *> \verbatim
00077 *>          AB is COMPLEX*16 array, dimension (LDAB,N)
00078 *>          On entry, the upper or lower triangle of the Hermitian band
00079 *>          matrix A, stored in the first kd+1 rows of the array.  The
00080 *>          j-th column of A is stored in the j-th column of the array AB
00081 *>          as follows:
00082 *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00083 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00084 *>
00085 *>          On exit, if INFO = 0, the factor S from the split Cholesky
00086 *>          factorization A = S**H*S. See Further Details.
00087 *> \endverbatim
00088 *>
00089 *> \param[in] LDAB
00090 *> \verbatim
00091 *>          LDAB is INTEGER
00092 *>          The leading dimension of the array AB.  LDAB >= KD+1.
00093 *> \endverbatim
00094 *>
00095 *> \param[out] INFO
00096 *> \verbatim
00097 *>          INFO is INTEGER
00098 *>          = 0: successful exit
00099 *>          < 0: if INFO = -i, the i-th argument had an illegal value
00100 *>          > 0: if INFO = i, the factorization could not be completed,
00101 *>               because the updated element a(i,i) was negative; the
00102 *>               matrix A is not positive definite.
00103 *> \endverbatim
00104 *
00105 *  Authors:
00106 *  ========
00107 *
00108 *> \author Univ. of Tennessee 
00109 *> \author Univ. of California Berkeley 
00110 *> \author Univ. of Colorado Denver 
00111 *> \author NAG Ltd. 
00112 *
00113 *> \date November 2011
00114 *
00115 *> \ingroup complex16OTHERcomputational
00116 *
00117 *> \par Further Details:
00118 *  =====================
00119 *>
00120 *> \verbatim
00121 *>
00122 *>  The band storage scheme is illustrated by the following example, when
00123 *>  N = 7, KD = 2:
00124 *>
00125 *>  S = ( s11  s12  s13                     )
00126 *>      (      s22  s23  s24                )
00127 *>      (           s33  s34                )
00128 *>      (                s44                )
00129 *>      (           s53  s54  s55           )
00130 *>      (                s64  s65  s66      )
00131 *>      (                     s75  s76  s77 )
00132 *>
00133 *>  If UPLO = 'U', the array AB holds:
00134 *>
00135 *>  on entry:                          on exit:
00136 *>
00137 *>   *    *   a13  a24  a35  a46  a57   *    *   s13  s24  s53**H s64**H s75**H
00138 *>   *   a12  a23  a34  a45  a56  a67   *   s12  s23  s34  s54**H s65**H s76**H
00139 *>  a11  a22  a33  a44  a55  a66  a77  s11  s22  s33  s44  s55    s66    s77
00140 *>
00141 *>  If UPLO = 'L', the array AB holds:
00142 *>
00143 *>  on entry:                          on exit:
00144 *>
00145 *>  a11  a22  a33  a44  a55  a66  a77  s11    s22    s33    s44  s55  s66  s77
00146 *>  a21  a32  a43  a54  a65  a76   *   s12**H s23**H s34**H s54  s65  s76   *
00147 *>  a31  a42  a53  a64  a64   *    *   s13**H s24**H s53    s64  s75   *    *
00148 *>
00149 *>  Array elements marked * are not used by the routine; s12**H denotes
00150 *>  conjg(s12); the diagonal elements of S are real.
00151 *> \endverbatim
00152 *>
00153 *  =====================================================================
00154       SUBROUTINE ZPBSTF( UPLO, N, KD, AB, LDAB, INFO )
00155 *
00156 *  -- LAPACK computational routine (version 3.4.0) --
00157 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00158 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00159 *     November 2011
00160 *
00161 *     .. Scalar Arguments ..
00162       CHARACTER          UPLO
00163       INTEGER            INFO, KD, LDAB, N
00164 *     ..
00165 *     .. Array Arguments ..
00166       COMPLEX*16         AB( LDAB, * )
00167 *     ..
00168 *
00169 *  =====================================================================
00170 *
00171 *     .. Parameters ..
00172       DOUBLE PRECISION   ONE, ZERO
00173       PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
00174 *     ..
00175 *     .. Local Scalars ..
00176       LOGICAL            UPPER
00177       INTEGER            J, KLD, KM, M
00178       DOUBLE PRECISION   AJJ
00179 *     ..
00180 *     .. External Functions ..
00181       LOGICAL            LSAME
00182       EXTERNAL           LSAME
00183 *     ..
00184 *     .. External Subroutines ..
00185       EXTERNAL           XERBLA, ZDSCAL, ZHER, ZLACGV
00186 *     ..
00187 *     .. Intrinsic Functions ..
00188       INTRINSIC          DBLE, MAX, MIN, SQRT
00189 *     ..
00190 *     .. Executable Statements ..
00191 *
00192 *     Test the input parameters.
00193 *
00194       INFO = 0
00195       UPPER = LSAME( UPLO, 'U' )
00196       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00197          INFO = -1
00198       ELSE IF( N.LT.0 ) THEN
00199          INFO = -2
00200       ELSE IF( KD.LT.0 ) THEN
00201          INFO = -3
00202       ELSE IF( LDAB.LT.KD+1 ) THEN
00203          INFO = -5
00204       END IF
00205       IF( INFO.NE.0 ) THEN
00206          CALL XERBLA( 'ZPBSTF', -INFO )
00207          RETURN
00208       END IF
00209 *
00210 *     Quick return if possible
00211 *
00212       IF( N.EQ.0 )
00213      $   RETURN
00214 *
00215       KLD = MAX( 1, LDAB-1 )
00216 *
00217 *     Set the splitting point m.
00218 *
00219       M = ( N+KD ) / 2
00220 *
00221       IF( UPPER ) THEN
00222 *
00223 *        Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
00224 *
00225          DO 10 J = N, M + 1, -1
00226 *
00227 *           Compute s(j,j) and test for non-positive-definiteness.
00228 *
00229             AJJ = DBLE( AB( KD+1, J ) )
00230             IF( AJJ.LE.ZERO ) THEN
00231                AB( KD+1, J ) = AJJ
00232                GO TO 50
00233             END IF
00234             AJJ = SQRT( AJJ )
00235             AB( KD+1, J ) = AJJ
00236             KM = MIN( J-1, KD )
00237 *
00238 *           Compute elements j-km:j-1 of the j-th column and update the
00239 *           the leading submatrix within the band.
00240 *
00241             CALL ZDSCAL( KM, ONE / AJJ, AB( KD+1-KM, J ), 1 )
00242             CALL ZHER( 'Upper', KM, -ONE, AB( KD+1-KM, J ), 1,
00243      $                 AB( KD+1, J-KM ), KLD )
00244    10    CONTINUE
00245 *
00246 *        Factorize the updated submatrix A(1:m,1:m) as U**H*U.
00247 *
00248          DO 20 J = 1, M
00249 *
00250 *           Compute s(j,j) and test for non-positive-definiteness.
00251 *
00252             AJJ = DBLE( AB( KD+1, J ) )
00253             IF( AJJ.LE.ZERO ) THEN
00254                AB( KD+1, J ) = AJJ
00255                GO TO 50
00256             END IF
00257             AJJ = SQRT( AJJ )
00258             AB( KD+1, J ) = AJJ
00259             KM = MIN( KD, M-J )
00260 *
00261 *           Compute elements j+1:j+km of the j-th row and update the
00262 *           trailing submatrix within the band.
00263 *
00264             IF( KM.GT.0 ) THEN
00265                CALL ZDSCAL( KM, ONE / AJJ, AB( KD, J+1 ), KLD )
00266                CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
00267                CALL ZHER( 'Upper', KM, -ONE, AB( KD, J+1 ), KLD,
00268      $                    AB( KD+1, J+1 ), KLD )
00269                CALL ZLACGV( KM, AB( KD, J+1 ), KLD )
00270             END IF
00271    20    CONTINUE
00272       ELSE
00273 *
00274 *        Factorize A(m+1:n,m+1:n) as L**H*L, and update A(1:m,1:m).
00275 *
00276          DO 30 J = N, M + 1, -1
00277 *
00278 *           Compute s(j,j) and test for non-positive-definiteness.
00279 *
00280             AJJ = DBLE( AB( 1, J ) )
00281             IF( AJJ.LE.ZERO ) THEN
00282                AB( 1, J ) = AJJ
00283                GO TO 50
00284             END IF
00285             AJJ = SQRT( AJJ )
00286             AB( 1, J ) = AJJ
00287             KM = MIN( J-1, KD )
00288 *
00289 *           Compute elements j-km:j-1 of the j-th row and update the
00290 *           trailing submatrix within the band.
00291 *
00292             CALL ZDSCAL( KM, ONE / AJJ, AB( KM+1, J-KM ), KLD )
00293             CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
00294             CALL ZHER( 'Lower', KM, -ONE, AB( KM+1, J-KM ), KLD,
00295      $                 AB( 1, J-KM ), KLD )
00296             CALL ZLACGV( KM, AB( KM+1, J-KM ), KLD )
00297    30    CONTINUE
00298 *
00299 *        Factorize the updated submatrix A(1:m,1:m) as U**H*U.
00300 *
00301          DO 40 J = 1, M
00302 *
00303 *           Compute s(j,j) and test for non-positive-definiteness.
00304 *
00305             AJJ = DBLE( AB( 1, J ) )
00306             IF( AJJ.LE.ZERO ) THEN
00307                AB( 1, J ) = AJJ
00308                GO TO 50
00309             END IF
00310             AJJ = SQRT( AJJ )
00311             AB( 1, J ) = AJJ
00312             KM = MIN( KD, M-J )
00313 *
00314 *           Compute elements j+1:j+km of the j-th column and update the
00315 *           trailing submatrix within the band.
00316 *
00317             IF( KM.GT.0 ) THEN
00318                CALL ZDSCAL( KM, ONE / AJJ, AB( 2, J ), 1 )
00319                CALL ZHER( 'Lower', KM, -ONE, AB( 2, J ), 1,
00320      $                    AB( 1, J+1 ), KLD )
00321             END IF
00322    40    CONTINUE
00323       END IF
00324       RETURN
00325 *
00326    50 CONTINUE
00327       INFO = J
00328       RETURN
00329 *
00330 *     End of ZPBSTF
00331 *
00332       END
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