zhico(3P)

NAME

zhico - compute the UDU factorization and condition number of a Hermitian matrix A. If the condition number is not needed then xHIFA is slightly faster. It is typical to follow a call to xHICO with a call to xHISL to solve Ax = b or to xHIDI to compute the determinant, inverse, and inertia of A.

SYNOPSIS

CALL ZHICO (ZA, LDA, N, IPIVOT, DRCOND, ZWORK)

CALL CHICO (CA, LDA, N, IPIVOT, SRCOND, CWORK)

void zhico(doublecomplex ∗za, long int lda, long int n,
long int ∗ipivit, double ∗rcond)

void chico(complex ∗ca, long int lda, long int n,
long int ∗ipivit, float ∗rcond)

ARGUMENTS

xAOn entry, the upper triangle of the matrix A.
On exit, a UDU factorization of the matrix A. The strict lower triangle of A is not referenced.

LDALeading dimension of the array A as specified in a dimension or type statement. LDA >= max(1,N).

NOrder of the matrix A. N >= 0.

IPIVOTOn exit, a vector of pivot indices.

xRCONDOn exit, an estimate of the reciprocal condition number of
A. 0.0 >= RCOND >= 1.0. As the value of RCOND gets smaller, operations with A such as solving Ax = b may become less stable. If RCOND satisfies RCOND + 1.0 = 1.0 then A may be singular to working precision.

xWORKScratch array with a dimension of N.

SAMPLE PROGRAM

      PROGRAM TEST
      IMPLICIT NONE
C
      INTEGER    LDA, N
      PARAMETER (N = 3)
      PARAMETER (LDA = 3)
C
      REAL       RCOND
      COMPLEX    A(LDA,N), B(N), WORK(N)
      INTEGER    ICOL, IPIVOT(N), IROW
C
      EXTERNAL   CHICO, CHISL
      INTRINSIC CONJG
C
C     Initialize the array A to store the matrix A shown below.
C     Initialize the array B to store the vector b shown below.
C
C          1    1+2i 1+2i         95-180i
C     A = 1+2i   6   -2+6i    b = 545-118i
C         1+2i -2+6i   11         865+ 62i
C
      DATA A / (1.0,0.0), (8E8,8E8), (8E8,8E8),
     $        (1.0,-2.0), (6.0,0.0), (8E8,8E8),
     $        (1.0,-2.0), (6.0,-2.0), (11.0,0.0) /
      DATA B / (95.0,-180.0), (545.0,-118.0), (865.0,62.0) /
C
      PRINT 1000
      DO 100, IROW = 1, N
        PRINT 1010, (CONJG(A(ICOL,IROW)), ICOL = 1, IROW),
     $              (A(IROW,ICOL), ICOL = IROW + 1, N)
100 CONTINUE
      PRINT 1020
      DO 110, IROW = 1, N
        PRINT 1010, (A(IROW,ICOL), ICOL = 1, N)
110 CONTINUE
      PRINT 1030
      PRINT 1040, B
      CALL CHICO (A, LDA, N, IPIVOT, RCOND, WORK)
      PRINT 1050, RCOND
      IF ((RCOND + 1.0) .EQ. 1.0) THEN
        PRINT 1060
      END IF
      CALL CHISL (A, LDA, N, IPIVOT, B)
      PRINT 1070
      PRINT 1040, B
C
1000 FORMAT (1X, ’A in full form:’)
1010 FORMAT (4(: 3X, ’(’, F4.1, ’,’, F4.1, ’)’))
1020 FORMAT (/1X, ’A in Hermitian form: (∗ in unused elements)’)
1030 FORMAT (/1X, ’b:’)
1040 FORMAT (3X, ’(’, F6.1, ’,’, F6.1, ’)’)
1050 FORMAT (/1X, ’Reciprocal condition number of A:’, F6.3)
1060 FORMAT (1X, ’A may be singular to working precision.’)
1070 FORMAT (/1X, ’A∗∗(-1) ∗ b:’)
C
      END

SAMPLE OUTPUT

A in full form:
   ( 1.0, 0.0)   ( 1.0,-2.0)   ( 1.0,-2.0)
   ( 1.0, 2.0)   ( 6.0, 0.0)   ( 6.0,-2.0)
   ( 1.0, 2.0)   ( 6.0, 2.0)   (11.0, 0.0)

A in Hermitian form: (∗ in unused elements)
   ( 1.0, 0.0)   ( 1.0,-2.0)   ( 1.0,-2.0)
   (∗∗∗∗,∗∗∗∗)   ( 6.0, 0.0)   ( 6.0,-2.0)
   (∗∗∗∗,∗∗∗∗)   (∗∗∗∗,∗∗∗∗)   (11.0, 0.0)

b:
   ( 95.0,-180.0)
   ( 545.0,-118.0)
   ( 865.0, 62.0)

Reciprocal condition number of A: 0.001

A∗∗(-1) ∗ b:
   (   5.0,   0.0)
   ( 26.0,   0.0)
   ( 64.0,   0.0)