SPFTRI computes the inverse of a real (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 A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


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

A

          A is REAL 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, the symmetric inverse of the original matrix, in the


          same storage format.

INFO

          INFO is INTEGER


          = 0:  successful exit


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


          > 0:  if INFO = i, the (i,i) element of the factor U or L is


                zero, and the inverse could not be computed.

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