ZPFTRS(3) solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF

SYNOPSIS

SUBROUTINE ZPFTRS(
TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )

    
CHARACTER TRANSR, UPLO

    
INTEGER INFO, LDB, N, NRHS

    
COMPLEX*16 A( 0: * ), B( LDB, * )

PURPOSE

ZPFTRS solves a system of linear equations A*X = B with a Hermitian positive definite matrix A using the Cholesky factorization A = U**H*U or A = L*L**H computed by ZPFTRF.

ARGUMENTS

TRANSR (input) CHARACTER
= 'N': The Normal TRANSR of RFP A is stored;
= 'C': The Conjugate-transpose TRANSR of RFP A is stored.
UPLO (input) CHARACTER

= 'U': Upper triangle of RFP A is stored;
= 'L': Lower triangle of RFP A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.
A (input) COMPLEX*16 array, dimension ( N*(N+1)/2 );
The triangular factor U or L from the Cholesky factorization of RFP A = U**H*U or RFP A = L*L**H, as computed by ZPFTRF. See note below for more details about RFP A.
B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the right hand side matrix B. On exit, the solution matrix X.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

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