SLATEC Routines --- DGBCO ---
*DECK DGBCO
SUBROUTINE DGBCO (ABD, LDA, N, ML, MU, IPVT, RCOND, Z)
C***BEGIN PROLOGUE DGBCO
C***PURPOSE Factor a band matrix by Gaussian elimination and
C estimate the condition number of the matrix.
C***LIBRARY SLATEC (LINPACK)
C***CATEGORY D2A2
C***TYPE DOUBLE PRECISION (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 DGBCO factors a double precision band matrix by Gaussian
C elimination and estimates the condition of the matrix.
C
C If RCOND is not needed, DGBFA is slightly faster.
C To solve A*X = B , follow DGBCO by DGBSL.
C To compute INVERSE(A)*C , follow DGBCO by DGBSL.
C To compute DETERMINANT(A) , follow DGBCO by DGBDI.
C
C On Entry
C
C ABD DOUBLE PRECISION(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 DOUBLE PRECISION
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 DOUBLE PRECISION(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 DASUM, DAXPY, DDOT, DGBFA, DSCAL
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 DGBCO