SPFTRS solves a system of linear equations A*X = B with a symmetric


 positive definite matrix A using the Cholesky factorization


 A = U**T*U or A = L*L**T computed by SPFTRF.

TRANSR

          TRANSR is CHARACTER*1


          = 'N':  The Normal TRANSR of RFP A is stored;


          = 'T':  The 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.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

A

          A is REAL 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**T, as computed by SPFTRF.


          See note below for more details about RFP A.

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the right hand side matrix B.


          On exit, the solution matrix X.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  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