# cgbco.f

```      SUBROUTINE CGBCO (ABD, LDA, N, ML, MU, IPVT, RCOND, Z)
C***BEGIN PROLOGUE  CGBCO
C***PURPOSE  Factor a band matrix by Gaussian elimination and
C            estimate the condition number of the matrix.
C***LIBRARY   SLATEC (LINPACK)
C***CATEGORY  D2C2
C***TYPE      COMPLEX (SGBCO-S, DGBCO-D, CGBCO-C)
C***KEYWORDS  BANDED, CONDITION NUMBER, LINEAR ALGEBRA, LINPACK,
C             MATRIX FACTORIZATION
C***AUTHOR  Moler, C. B., (U. of New Mexico)
C***DESCRIPTION
C
C     CGBCO factors a complex band matrix by Gaussian
C     elimination and estimates the condition of the matrix.
C
C     If  RCOND  is not needed, CGBFA is slightly faster.
C     To solve  A*X = B , follow CGBCO by CGBSL.
C     To compute  INVERSE(A)*C , follow CGBCO by CGBSL.
C     To compute  DETERMINANT(A) , follow CGBCO by CGBDI.
C
C     On Entry
C
C        ABD     COMPLEX(LDA, N)
C                contains the matrix in band storage.  The columns
C                of the matrix are stored in the columns of  ABD  and
C                the diagonals of the matrix are stored in rows
C                ML+1 through 2*ML+MU+1 of  ABD .
C                See the comments below for details.
C
C        LDA     INTEGER
C                the leading dimension of the array  ABD .
C                LDA must be .GE. 2*ML + MU + 1 .
C
C        N       INTEGER
C                the order of the original matrix.
C
C        ML      INTEGER
C                number of diagonals below the main diagonal.
C                0 .LE. ML .LT. N .
C
C        MU      INTEGER
C                number of diagonals above the main diagonal.
C                0 .LE. MU .LT. N .
C                More efficient if  ML .LE. MU .
C
C     On Return
C
C        ABD     an upper triangular matrix in band storage and
C                the multipliers which were used to obtain it.
C                The factorization can be written  A = L*U  where
C                L  is a product of permutation and unit lower
C                triangular matrices and  U  is upper triangular.
C
C        IPVT    INTEGER(N)
C                an integer vector of pivot indices.
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.
C
C        Z       COMPLEX(N)
C                a work vector whose contents are usually unimportant.
C                If  A  is close to a singular matrix, then  Z  is
C                an approximate null vector in the sense that
C                NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
C
C     Band Storage
C
C           if  A  is a band matrix, the following program segment
C           will set up the input.
C
C                   ML = (band width below the diagonal)
C                   MU = (band width above the diagonal)
C                   M = ML + MU + 1
C                   DO 20 J = 1, N
C                      I1 = MAX(1, J-MU)
C                      I2 = MIN(N, J+Ml)
C                      DO 10 I = I1, I2
C                         K = I - J + M
C                         ABD(K,J) = A(I,J)
C                10    CONTINUE
C                20 CONTINUE
C
C           This uses rows  ML+1  through  2*ML+MU+1  of  ABD .
C           In addition, the first  ML  rows in  ABD  are used for
C           elements generated during the triangularization.
C           The total number of rows needed in  ABD  is  2*ML+MU+1 .
C           The  ML+MU by ML+MU  upper left triangle and the
C           ML by ML  lower right triangle are not referenced.
C
C     Example:  If the original matrix is
C
C           11 12 13  0  0  0
C           21 22 23 24  0  0
C            0 32 33 34 35  0
C            0  0 43 44 45 46
C            0  0  0 54 55 56
C            0  0  0  0 65 66
C
C      then  N = 6, ML = 1, MU = 2, LDA .GE. 5  and ABD should contain
C
C            *  *  *  +  +  +  , * = not used
C            *  * 13 24 35 46  , + = used for pivoting
C            * 12 23 34 45 56
C           11 22 33 44 55 66
C           21 32 43 54 65  *
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  CAXPY, CDOTC, CGBFA, CSSCAL, SCASUM
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  CGBCO
```