SLATEC Routines --- DPBCO ---
*DECK DPBCO
SUBROUTINE DPBCO (ABD, LDA, N, M, RCOND, Z, INFO)
C***BEGIN PROLOGUE DPBCO
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 DOUBLE 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 DPBCO factors a double precision symmetric positive definite
C matrix stored in band form and estimates the condition of the
C matrix.
C
C If RCOND is not needed, DPBFA is slightly faster.
C To solve A*X = B , follow DPBCO by DPBSL.
C To compute INVERSE(A)*C , follow DPBCO by DPBSL.
C To compute DETERMINANT(A) , follow DPBCO by DPBDI.
C
C On Entry
C
C ABD DOUBLE PRECISION(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 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. If INFO .NE. 0 , RCOND is unchanged.
C
C Z DOUBLE PRECISION(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 DASUM, DAXPY, DDOT, DPBFA, 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 DPBCO