DPFTRF(3)      LAPACK routine of NEC Numeric Library Collection      DPFTRF(3)



NAME
       DPFTRF

SYNOPSIS
       SUBROUTINE DPFTRF (TRANSR, UPLO, N, A, INFO)



PURPOSE
            DPFTRF computes the Cholesky factorization of a real symmetric
            positive definite matrix A.

            The factorization has the form
               A = U**T * U,  if UPLO = 'U', or
               A = L  * L**T,  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.




ARGUMENTS
           TRANSR    (input)
                     TRANSR is CHARACTER*1
                     = 'N':  The Normal TRANSR of RFP A is stored;
                     = 'T':  The Transpose TRANSR of RFP A is stored.

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

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

           A         (input/output)
                     A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
                     On entry, the symmetric 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 = 'T' then RFP is
                     the 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 =
                     'T'. 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**T*U or RFP A = L*L**T.

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






FURTHER DETAILS
             We first consider Rectangular Full Packed (RFP) 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
             the 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
             the transpose of the last three columns of AP lower.
             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 = 'T'. RFP A in both UPLO cases is just the
             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 then consider Rectangular Full Packed (RFP) 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
             the 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
             the transpose of the last two columns of AP lower.
             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 = 'T'. RFP A in both UPLO cases is just the
             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



LAPACK routine                  31 October 2017                      DPFTRF(3)