SGBCO(3F)

NAME
     SGBCO   - SBGCO factors a real band matrix by Gaussian elimination and
     estimates the condition of the matrix.

     If  RCOND  is not needed, SGBFA is slightly faster.  To solve  A*X = B ,
     follow SBGCO by SGBSL.  To compute  INVERSE(A)*C , follow SBGCO by SGBSL.
     To compute  DETERMINANT(A) , follow SBGCO by SGBDI.


SYNOPSYS
      SUBROUTINE SGBCO(ABD,LDA,N,ML,MU,IPVT,RCOND,Z)

DESCRIPTION
     On Entry

     ABD REAL(LDA, N)
        contains the matrix in band storage.  The columns
        of the matrix are stored in the columns of  ABD  and
        the diagonals of the matrix are stored in rows
        ML+1 through 2*ML+MU+1 of  ABD .
        See the comments below for details.

     LDA INTEGER
        the leading dimension of the array  ABD .
        LDA must be .GE. 2*ML + MU + 1 .

     N INTEGER
        the order of the original matrix.

     ML INTEGER
        number of diagonals below the main diagonal.
        0 .LE. ML .LT. N .

     MU INTEGER
        number of diagonals above the main diagonal.
        0 .LE. MU .LT. N .
        More efficient if  ML .LE. MU .  On Return

     ABD an upper triangular matrix in band storage and
        the multipliers which were used to obtain it.
        The factorization can be written  A = L*U  where
        L  is a product of permutation and unit lower
        triangular matrices and  U  is upper triangular.

     IPVT INTEGER(N)
        an integer vector of pivot indices.

     RCOND REAL
        an estimate of the reciprocal condition of  A .
        For the system  A*X = B , relative perturbations
        in  A  and  B  of size  EPSILON  may cause
        relative perturbations in  X  of size  EPSILON/RCOND .
        If  RCOND  is so small that the logical expression
        1.0 + RCOND .EQ. 1.0
        is true, then  A  may be singular to working
        precision.  In particular,  RCOND  is zero  if
        exact singularity is detected or the estimate
        underflows.

     Z REAL(N)
        a work vector whose contents are usually unimportant.
        If  A  is close to a singular matrix, then  Z  is
        an approximate null vector in the sense that
        NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .  Band Storage
        If  A  is a band matrix, the following program segment
        will set up the input.
        ML = (band width below the diagonal)
        MU = (band width above the diagonal)
        M = ML + MU + 1
        DO 20 J = 1, N
        I1 = MAX0(1, J-MU)
        I2 = MIN0(N, J+ML)
        DO 10 I = I1, I2
        K = I - J + M
        ABD(K,J) = A(I,J)
        10    CONTINUE
        20 CONTINUE
        This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
        In addition, the first  ML  rows in  ABD  are used for
        elements generated during the triangularization.
        The total number of rows needed in  ABD  is  2*ML+MU+1 .
        The  ML+MU by ML+MU  upper left triangle and the
        ML by ML  lower right triangle are not referenced.  Example:  If the
     original matrix is
        11 12 13  0  0  0
        21 22 23 24  0  0
        0 32 33 34 35  0
        0  0 43 44 45 46
        0  0  0 54 55 56
        0  0  0  0 65 66 then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABD
     should contain
        *  *  *  +  +  +  , * = not used
        *  * 13 24 35 46  , + = used for pivoting
        * 12 23 34 45 56
        11 22 33 44 55 66
        21 32 43 54 65  * LINPACK.  This version dated 08/14/78 .  Cleve
     Moler, University of New Mexico, Argonne National Lab.  Subroutines and
     Functions LINPACK SGBFA BLAS SAXPY,SDOT,SSCAL,SASUM Fortran
     ABS,AMAX1,MAX0,MIN0,SIGN