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NAME
     schdd - downdate an augmented Cholesky decomposition of  the
     triangular part of an augmented QR decomposition.

SYNOPSIS
     SUBROUTINE DCHDD (DA, LDA, N, DX, DZ,  LDZ,  NZ,  DY,  DRHO,
               DCOS, DSIN, INFO)

     SUBROUTINE SCHDD (SA, LDA, N, SX, SZ,  LDZ,  NZ,  SY,  SRHO,
               SCOS, SSIN, INFO)

     SUBROUTINE ZCHDD (ZA, LDA, N, ZX, ZZ,  LDZ,  NZ,  ZY,  DRHO,
               DCOS, DSIN, INFO)

     SUBROUTINE CCHDD (CA, LDA, N, CX, CZ,  LDZ,  NZ,  CY,  SRHO,
               SCOS, SSIN, INFO)



     #include <sunperf.h>

     void dchdd(double *r, int ldr, int p, double *dx, double *z,
               int  ldz, int nz, double *dy, double *drho, double
               *c, double *s, int *info)

     void schdd(float *r, int ldr, int p, float  *sx,  float  *z,
               int  ldz,  int  nz,  float *sy, float *srho, float
               *sc, float *s, int *info)

     void zchdd(doublecomplex *r, int ldr, int  p,  doublecomplex
               *x, doublecomplex *zz, int ldz, int nz, doublecom-
               plex *y, doublecomplex *zrho,  doublecomplex  *zc,
               doublecomplex *s, int *info)

     void cchdd(complex *r, int ldr, int p, complex *cx,  complex
               *cz,  int  ldz, int nz, complex *cy, complex *rho,
               complex *cc, complex *s, int *info) ;

ARGUMENTS
     xA        On entry, the upper triangular matrix A.  On exit,
               A  has  been downdated.  The strict lower triangle
               of A is not referenced.

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

     N         Order of the matrix A.  N >= 0.

     xX        Row to be added to A.

     xZ        Vectors to be downdated with A.

     LDZ       Leading dimension on the array Z as specified in a
               dimension or type statement.  LDZ >= max(1,N).

     NZ        Number of vectors to be downdated with A.   NZ  >=
               0.  If NZ = 0 then Z, Y, and RHO are not used.

     xY        Scalars for downdating the vectors in Z.

     xRHO      On entry, the norms of the residual  vectors  that
               are  to be downdated.  On exit, RHO has been down-
               dated.  If RHO(i) is negative on entry then it  is
               not changed.

     xCOS      Cosines of the transforming rotations.

     xSIN      Sines of the transforming rotations.

     INFO      On exit:
               INFO = 0  Subroutine completed normally.
               INFO = -1 A could not be downdated; all values are
               left unchanged.
               INFO = 1  Some RHOs could not  be  downdated;  all
               RHOs  that  could  not be downdated are changed to
               -1.

SAMPLE PROGRAM
           PROGRAM TEST
           IMPLICIT NONE
     C
           INTEGER           LDA, N, NOPIV, NZ
           PARAMETER        (N = 4)
           PARAMETER        (LDA = N)
           PARAMETER        (NOPIV = 0)
           PARAMETER        (NZ = 0)
     C
           DOUBLE PRECISION  A(LDA,N), ANULL, C(N), S(N), WORK(N), X(N)
           INTEGER           I, INFO, IPIVOT(N), J, JOB, NULL
     C
           EXTERNAL          DCHDC, DCHDD
     C
     C     Initialize the arrays A and Z to store the matrices A and Z
     C     shown below and initialize X and Y to store the vectors x and y
     C     shown below.
     C
     C         4  3  2  1        1
     C     A = 3  4  3  2    x = 1
     C         2  3  4  3        1
     C         1  2  3  4        1
     C
           DATA A / 4.0D0, 3*8D8, 3.0D0, 4.0D0, 2*8D8, 2.0D0, 3.0D0, 4.0D0,
          $         8D8, 1.0D0, 2.0D0, 3.0D0, 4.0D0 /
     C
           PRINT 1000
           DO 100, I = 1, N
             PRINT 1010, (A(J,I), J = 1, I), (A(I,J), J = I + 1, N)
       100 CONTINUE
           PRINT 1020
           PRINT 1010, ((A(I,J), J = 1, N), I = 1, N)
           JOB = NOPIV
           CALL DCHDC (A, LDA, N, WORK, IPIVOT, JOB, INFO)
           IF (INFO .EQ. N) THEN
             PRINT 1030
             PRINT 1010, A(1,1), A(1,2), A(1,3), A(1,4)
             PRINT 1040,         A(2,2), A(2,3), A(2,4)
             PRINT 1050,                 A(3,3), A(3,4)
             PRINT 1060,                         A(4,4)
             ANULL = 0.0D0
             NULL = 1
             CALL DCHDD (A, LDA, N, X, ANULL, NULL, NZ, ANULL, ANULL, C, S,
          $              INFO)
             IF (INFO .EQ. 0) THEN
               PRINT 1070
               PRINT 1080, (C(I), S(I), I = 1, N)
             ELSE
               PRINT 1090
             END IF
           ELSE
             PRINT 1100
           END IF
     C
      1000 FORMAT (1X, 'A in full form:')
      1010 FORMAT (4(3X, F7.3))
      1020 FORMAT (/1X, 'A in symmetric form (* in unused entries)')
      1030 FORMAT (/1X, 'Upper Cholesky factor:')
      1040 FORMAT (10X, 3(3X, F7.3))
      1050 FORMAT (20X, 2(3X, F7.3))
      1060 FORMAT (30X, 1(3X, F7.3))
      1070 FORMAT (/1X, 'Cosine', 3X, '  Sine')
      1080 FORMAT (1X, F6.3, 3X, F6.3)
      1090 FORMAT (/1X, 'A cannot be downdated.')
      1100 FORMAT (/1X, 'A is not positive definite.')
     C
           END

SAMPLE OUTPUT
      A in full form:
          4.000     3.000     2.000     1.000
          3.000     4.000     3.000     2.000
          2.000     3.000     4.000     3.000
          1.000     2.000     3.000     4.000

      A in symmetric form (* in unused entries)
          4.000     3.000     2.000     1.000
        *******     4.000     3.000     2.000
        *******   *******     4.000     3.000
        *******   *******   *******     4.000

      Upper Cholesky factor:
          2.000     1.500     1.000     0.500
                    1.323     1.134     0.945
                              1.309     1.091
                                        1.291

      Cosine     Sine
       1.000    0.000
       1.000    0.000
       1.000    0.000
       1.000    0.000

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