cpftrf.f - Man Page

subroutine cpftrf (transr, uplo, n, a, info)
CPFTRF

CPFTRF  

Purpose:

 CPFTRF computes the Cholesky factorization of a complex Hermitian
 positive definite matrix A.

 The factorization has the form
    A = U**H * U,  if UPLO = 'U', or
    A = L  * L**H,  if UPLO = 'L',
 where U is an upper triangular matrix and L is lower triangular.

 This is the block version of the algorithm, calling Level 3 BLAS.

Parameters

TRANSR

          TRANSR is CHARACTER*1
          = 'N':  The Normal TRANSR of RFP A is stored;
          = 'C':  The Conjugate-transpose TRANSR of RFP A is stored.

UPLO

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of RFP A is stored;
          = 'L':  Lower triangle of RFP A is stored.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension ( N*(N+1)/2 );
          On entry, the Hermitian matrix A in RFP format. RFP format is
          described by TRANSR, UPLO, and N as follows: If TRANSR = 'N'
          then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is
          (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is
          the Conjugate-transpose of RFP A as defined when
          TRANSR = 'N'. The contents of RFP A are defined by UPLO as
          follows: If UPLO = 'U' the RFP A contains the nt elements of
          upper packed A. If UPLO = 'L' the RFP A contains the elements
          of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR =
          'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N
          is odd. See the Note below for more details.

          On exit, if INFO = 0, the factor U or L from the Cholesky
          factorization RFP A = U**H*U or RFP A = L*L**H.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
          > 0:  if INFO = i, the leading principal minor of order i
                is not positive, and the factorization could not be
                completed.

  Further Notes on RFP Format:
  ============================

  We first consider Standard Packed Format when N is even.
  We give an example where N = 6.

     AP is Upper             AP is Lower

   00 01 02 03 04 05       00
      11 12 13 14 15       10 11
         22 23 24 25       20 21 22
            33 34 35       30 31 32 33
               44 45       40 41 42 43 44
                  55       50 51 52 53 54 55

  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
  three columns of AP upper. The lower triangle A(4:6,0:2) consists of
  conjugate-transpose of the first three columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:2,0:2) consists of
  conjugate-transpose of the last three columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N even and TRANSR = 'N'.

         RFP A                   RFP A

                                -- -- --
        03 04 05                33 43 53
                                   -- --
        13 14 15                00 44 54
                                      --
        23 24 25                10 11 55

        33 34 35                20 21 22
        --
        00 44 45                30 31 32
        -- --
        01 11 55                40 41 42
        -- -- --
        02 12 22                50 51 52

  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:

           RFP A                   RFP A

     -- -- -- --                -- -- -- -- -- --
     03 13 23 33 00 01 02    33 00 10 20 30 40 50
     -- -- -- -- --                -- -- -- -- --
     04 14 24 34 44 11 12    43 44 11 21 31 41 51
     -- -- -- -- -- --                -- -- -- --
     05 15 25 35 45 55 22    53 54 55 22 32 42 52

  We next  consider Standard Packed Format when N is odd.
  We give an example where N = 5.

     AP is Upper                 AP is Lower

   00 01 02 03 04              00
      11 12 13 14              10 11
         22 23 24              20 21 22
            33 34              30 31 32 33
               44              40 41 42 43 44

  Let TRANSR = 'N'. RFP holds AP as follows:
  For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
  three columns of AP upper. The lower triangle A(3:4,0:1) consists of
  conjugate-transpose of the first two   columns of AP upper.
  For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
  three columns of AP lower. The upper triangle A(0:1,1:2) consists of
  conjugate-transpose of the last two   columns of AP lower.
  To denote conjugate we place -- above the element. This covers the
  case N odd  and TRANSR = 'N'.

         RFP A                   RFP A

                                   -- --
        02 03 04                00 33 43
                                      --
        12 13 14                10 11 44

        22 23 24                20 21 22
        --
        00 33 34                30 31 32
        -- --
        01 11 44                40 41 42

  Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
  transpose of RFP A above. One therefore gets:

           RFP A                   RFP A

     -- -- --                   -- -- -- -- -- --
     02 12 22 00 01             00 10 20 30 40 50
     -- -- -- --                   -- -- -- -- --
     03 13 23 33 11             33 11 21 31 41 51
     -- -- -- -- --                   -- -- -- --
     04 14 24 34 44             43 44 22 32 42 52

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 210 of file cpftrf.f.