# spbco.f

          SUBROUTINE SPBCO (ABD, LDA, N, M, RCOND, Z, INFO)
    C***BEGIN PROLOGUE  SPBCO
    C***PURPOSE  Factor a real symmetric positive definite matrix stored in
    C            band form and estimate the condition number of the matrix.
    C***LIBRARY   SLATEC (LINPACK)
    C***CATEGORY  D2B2
    C***TYPE      SINGLE PRECISION (SPBCO-S, DPBCO-D, CPBCO-C)
    C***KEYWORDS  BANDED, CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
    C             MATRIX FACTORIZATION, POSITIVE DEFINITE
    C***AUTHOR  Moler, C. B., (U. of New Mexico)
    C***DESCRIPTION
    C
    C     SPBCO factors a real symmetric positive definite matrix
    C     stored in band form and estimates the condition of the matrix.
    C
    C     If  RCOND  is not needed, SPBFA is slightly faster.
    C     To solve  A*X = B , follow SPBCO by SPBSL.
    C     To compute  INVERSE(A)*C , follow SPBCO by SPBSL.
    C     To compute  DETERMINANT(A) , follow SPBCO by SPBDI.
    C
    C     On Entry
    C
    C        ABD     REAL(LDA, N)
    C                the matrix to be factored.  The columns of the upper
    C                triangle are stored in the columns of ABD and the
    C                diagonals of the upper triangle are stored in the
    C                rows of ABD .  See the comments below for details.
    C
    C        LDA     INTEGER
    C                the leading dimension of the array  ABD .
    C                LDA must be .GE. M + 1 .
    C
    C        N       INTEGER
    C                the order of the matrix  A .
    C
    C        M       INTEGER
    C                the number of diagonals above the main diagonal.
    C                0 .LE. M .LT. N .
    C
    C     On Return
    C
    C        ABD     an upper triangular matrix  R , stored in band
    C                form, so that  A = TRANS(R)*R .
    C                If  INFO .NE. 0 , the factorization is not complete.
    C
    C        RCOND   REAL
    C                an estimate of the reciprocal condition of  A .
    C                For the system  A*X = B , relative perturbations
    C                in  A  and  B  of size  EPSILON  may cause
    C                relative perturbations in  X  of size  EPSILON/RCOND .
    C                If  RCOND  is so small that the logical expression
    C                           1.0 + RCOND .EQ. 1.0
    C                is true, then  A  may be singular to working
    C                precision.  In particular,  RCOND  is zero  if
    C                exact singularity is detected or the estimate
    C                underflows.  If INFO .NE. 0 , RCOND is unchanged.
    C
    C        Z       REAL(N)
    C                a work vector whose contents are usually unimportant.
    C                If  A  is singular to working precision, then  Z  is
    C                an approximate null vector in the sense that
    C                NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
    C                If  INFO .NE. 0 , Z  is unchanged.
    C
    C        INFO    INTEGER
    C                = 0  for normal return.
    C                = K  signals an error condition.  The leading minor
    C                     of order  K  is not positive definite.
    C
    C     Band Storage
    C
    C           If  A  is a symmetric positive definite band matrix,
    C           the following program segment will set up the input.
    C
    C                   M = (band width above diagonal)
    C                   DO 20 J = 1, N
    C                      I1 = MAX(1, J-M)
    C                      DO 10 I = I1, J
    C                         K = I-J+M+1
    C                         ABD(K,J) = A(I,J)
    C                10    CONTINUE
    C                20 CONTINUE
    C
    C           This uses  M + 1  rows of  A , except for the  M by M
    C           upper left triangle, which is ignored.
    C
    C     Example:  If the original matrix is
    C
    C           11 12 13  0  0  0
    C           12 22 23 24  0  0
    C           13 23 33 34 35  0
    C            0 24 34 44 45 46
    C            0  0 35 45 55 56
    C            0  0  0 46 56 66
    C
    C     then  N = 6 , M = 2  and  ABD  should contain
    C
    C            *  * 13 24 35 46
    C            * 12 23 34 45 56
    C           11 22 33 44 55 66
    C
    C***REFERENCES  J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
    C                 Stewart, LINPACK Users' Guide, SIAM, 1979.
    C***ROUTINES CALLED  SASUM, SAXPY, SDOT, SPBFA, SSCAL
    C***REVISION HISTORY  (YYMMDD)
    C   780814  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   890831  Modified array declarations.  (WRB)
    C   890831  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   900326  Removed duplicate information from DESCRIPTION section.
    C           (WRB)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  SPBCO