DPBCO(3F)							     DPBCO(3F)

NAME    [Toc]    [Back]

     DPBCO   - DPBCO factors a double precision	symmetric positive definite
     matrix stored in band form	and estimates the condition of the matrix.

     If	 RCOND	is not needed, DPBFA is	slightly faster.  To solve  A*X	= B ,
     follow DPBCO by DPBSL.  To	compute	 INVERSE(A)*C ,	follow DPBCO by	DPBSL.
     To	compute	 DETERMINANT(A)	, follow DPBCO by DPBDI.

SYNOPSYS    [Toc]    [Back]

      SUBROUTINE DPBCO(ABD,LDA,N,M,RCOND,Z,INFO)

DESCRIPTION    [Toc]    [Back]

     On	Entry

     ABD DOUBLE	PRECISION(LDA, N)
	the matrix to be factored.  The	columns	of the upper
	triangle are stored in the columns of ABD and the
	diagonals of the upper triangle	are stored in the
	rows of	ABD .  See the comments	below for details.

     LDA INTEGER
	the leading dimension of the array  ABD	.
	LDA must be .GE. M + 1 .

     N INTEGER
	the order of the matrix	 A .

     M INTEGER
	the number of diagonals	above the main diagonal.
	0 .LE. M .LT.  N .  On Return

     ABD an upper triangular matrix  R , stored	in band
	form, so that  A = TRANS(R)*R .
	If  INFO .NE. 0	, the factorization is not complete.

     RCOND DOUBLE PRECISION
	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.  If	INFO .NE. 0 , RCOND is unchanged.

     Z DOUBLE PRECISION(N)
	a work vector whose contents are usually unimportant.
	If  A  is singular to working precision, then  Z  is
	an approximate null vector in the sense	that



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DPBCO(3F)							     DPBCO(3F)



	NORM(A*Z) = RCOND*NORM(A)*NORM(Z) .
	If  INFO .NE. 0	, Z  is	unchanged.

     INFO INTEGER
	= 0  for normal	return.
	= K  signals an	error condition.  The leading minor
	of order  K  is	not positive definite.	Band Storage
	If  A  is a symmetric positive definite	band matrix,
	the following program segment will set up the input.
	M = (band width	above diagonal)
	DO 20 J	= 1, N
	I1 = MAX0(1, J-M)
	DO 10 I	= I1, J
	K = I-J+M+1
	ABD(K,J) = A(I,J)
	10    CONTINUE
	20 CONTINUE
	This uses  M + 1  rows of  A , except for the  M by M
	upper left triangle, which is ignored.	Example:  If the original
     matrix is
	11 12 13  0  0	0
	12 22 23 24  0	0
	13 23 33 34 35	0
	0 24 34	44 45 46
	0  0 35	45 55 56
	0  0  0	46 56 66 then  N = 6 , M = 2  and  ABD	should contain
	*  * 13	24 35 46
	* 12 23	34 45 56
	11 22 33 44 55 66 LINPACK.  This version dated 08/14/78	.  Cleve
     Moler, University of New Mexico, Argonne National Lab.  Subroutines and
     Functions LINPACK DPBFA BLAS DAXPY,DDOT,DSCAL,DASUM Fortran
     DABS,DMAX1,MAX0,MIN0,DREAL,DSIGN


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