ZPFTRS - Online Linux Manual Page

Section : 1
Updated : November 2008
Source : LAPACK routine (version 3.2)
Note : LAPACK routine (version 3.2)
NAMEZPFTRS - 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
SYNOPSISSUBROUTINE ZPFTRS(  TRANSR, UPLO, N, NRHS, A, B, LDB, INFO )  CHARACTER TRANSR, UPLO  INTEGER INFO, LDB, N, NRHS  COMPLEX*16 A( 0: * ), B( LDB, * )
PURPOSEZPFTRS 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.
ARGUMENTSTRANSR (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 DETAILSWe 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
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