LAPACK 3.3.0

spbtf2.f

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00001       SUBROUTINE SPBTF2( UPLO, N, KD, AB, LDAB, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.2) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *     November 2006
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          UPLO
00010       INTEGER            INFO, KD, LDAB, N
00011 *     ..
00012 *     .. Array Arguments ..
00013       REAL               AB( LDAB, * )
00014 *     ..
00015 *
00016 *  Purpose
00017 *  =======
00018 *
00019 *  SPBTF2 computes the Cholesky factorization of a real symmetric
00020 *  positive definite band matrix A.
00021 *
00022 *  The factorization has the form
00023 *     A = U' * U ,  if UPLO = 'U', or
00024 *     A = L  * L',  if UPLO = 'L',
00025 *  where U is an upper triangular matrix, U' is the transpose of U, and
00026 *  L is lower triangular.
00027 *
00028 *  This is the unblocked version of the algorithm, calling Level 2 BLAS.
00029 *
00030 *  Arguments
00031 *  =========
00032 *
00033 *  UPLO    (input) CHARACTER*1
00034 *          Specifies whether the upper or lower triangular part of the
00035 *          symmetric matrix A is stored:
00036 *          = 'U':  Upper triangular
00037 *          = 'L':  Lower triangular
00038 *
00039 *  N       (input) INTEGER
00040 *          The order of the matrix A.  N >= 0.
00041 *
00042 *  KD      (input) INTEGER
00043 *          The number of super-diagonals of the matrix A if UPLO = 'U',
00044 *          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
00045 *
00046 *  AB      (input/output) REAL array, dimension (LDAB,N)
00047 *          On entry, the upper or lower triangle of the symmetric band
00048 *          matrix A, stored in the first KD+1 rows of the array.  The
00049 *          j-th column of A is stored in the j-th column of the array AB
00050 *          as follows:
00051 *          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00052 *          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00053 *
00054 *          On exit, if INFO = 0, the triangular factor U or L from the
00055 *          Cholesky factorization A = U'*U or A = L*L' of the band
00056 *          matrix A, in the same storage format as A.
00057 *
00058 *  LDAB    (input) INTEGER
00059 *          The leading dimension of the array AB.  LDAB >= KD+1.
00060 *
00061 *  INFO    (output) INTEGER
00062 *          = 0: successful exit
00063 *          < 0: if INFO = -k, the k-th argument had an illegal value
00064 *          > 0: if INFO = k, the leading minor of order k is not
00065 *               positive definite, and the factorization could not be
00066 *               completed.
00067 *
00068 *  Further Details
00069 *  ===============
00070 *
00071 *  The band storage scheme is illustrated by the following example, when
00072 *  N = 6, KD = 2, and UPLO = 'U':
00073 *
00074 *  On entry:                       On exit:
00075 *
00076 *      *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
00077 *      *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
00078 *     a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
00079 *
00080 *  Similarly, if UPLO = 'L' the format of A is as follows:
00081 *
00082 *  On entry:                       On exit:
00083 *
00084 *     a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
00085 *     a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
00086 *     a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
00087 *
00088 *  Array elements marked * are not used by the routine.
00089 *
00090 *  =====================================================================
00091 *
00092 *     .. Parameters ..
00093       REAL               ONE, ZERO
00094       PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
00095 *     ..
00096 *     .. Local Scalars ..
00097       LOGICAL            UPPER
00098       INTEGER            J, KLD, KN
00099       REAL               AJJ
00100 *     ..
00101 *     .. External Functions ..
00102       LOGICAL            LSAME
00103       EXTERNAL           LSAME
00104 *     ..
00105 *     .. External Subroutines ..
00106       EXTERNAL           SSCAL, SSYR, XERBLA
00107 *     ..
00108 *     .. Intrinsic Functions ..
00109       INTRINSIC          MAX, MIN, SQRT
00110 *     ..
00111 *     .. Executable Statements ..
00112 *
00113 *     Test the input parameters.
00114 *
00115       INFO = 0
00116       UPPER = LSAME( UPLO, 'U' )
00117       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00118          INFO = -1
00119       ELSE IF( N.LT.0 ) THEN
00120          INFO = -2
00121       ELSE IF( KD.LT.0 ) THEN
00122          INFO = -3
00123       ELSE IF( LDAB.LT.KD+1 ) THEN
00124          INFO = -5
00125       END IF
00126       IF( INFO.NE.0 ) THEN
00127          CALL XERBLA( 'SPBTF2', -INFO )
00128          RETURN
00129       END IF
00130 *
00131 *     Quick return if possible
00132 *
00133       IF( N.EQ.0 )
00134      $   RETURN
00135 *
00136       KLD = MAX( 1, LDAB-1 )
00137 *
00138       IF( UPPER ) THEN
00139 *
00140 *        Compute the Cholesky factorization A = U'*U.
00141 *
00142          DO 10 J = 1, N
00143 *
00144 *           Compute U(J,J) and test for non-positive-definiteness.
00145 *
00146             AJJ = AB( KD+1, J )
00147             IF( AJJ.LE.ZERO )
00148      $         GO TO 30
00149             AJJ = SQRT( AJJ )
00150             AB( KD+1, J ) = AJJ
00151 *
00152 *           Compute elements J+1:J+KN of row J and update the
00153 *           trailing submatrix within the band.
00154 *
00155             KN = MIN( KD, N-J )
00156             IF( KN.GT.0 ) THEN
00157                CALL SSCAL( KN, ONE / AJJ, AB( KD, J+1 ), KLD )
00158                CALL SSYR( 'Upper', KN, -ONE, AB( KD, J+1 ), KLD,
00159      $                    AB( KD+1, J+1 ), KLD )
00160             END IF
00161    10    CONTINUE
00162       ELSE
00163 *
00164 *        Compute the Cholesky factorization A = L*L'.
00165 *
00166          DO 20 J = 1, N
00167 *
00168 *           Compute L(J,J) and test for non-positive-definiteness.
00169 *
00170             AJJ = AB( 1, J )
00171             IF( AJJ.LE.ZERO )
00172      $         GO TO 30
00173             AJJ = SQRT( AJJ )
00174             AB( 1, J ) = AJJ
00175 *
00176 *           Compute elements J+1:J+KN of column J and update the
00177 *           trailing submatrix within the band.
00178 *
00179             KN = MIN( KD, N-J )
00180             IF( KN.GT.0 ) THEN
00181                CALL SSCAL( KN, ONE / AJJ, AB( 2, J ), 1 )
00182                CALL SSYR( 'Lower', KN, -ONE, AB( 2, J ), 1,
00183      $                    AB( 1, J+1 ), KLD )
00184             END IF
00185    20    CONTINUE
00186       END IF
00187       RETURN
00188 *
00189    30 CONTINUE
00190       INFO = J
00191       RETURN
00192 *
00193 *     End of SPBTF2
00194 *
00195       END
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