subroutine spftri	(	character	TRANSR,
		character	UPLO,
		integer	N,
		real, dimension( 0: * )	A,
		integer	INFO
	)

SPFTRI

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Purpose:

 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.

Parameters

[in]	TRANSR	TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'T': The Transpose TRANSR of RFP A is stored.
[in]	UPLO	UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.
[in]	N	N is INTEGER The order of the matrix A. N >= 0.
[in,out]	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.
[out]	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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

November 2011

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

Definition at line 193 of file spftri.f.

*
*  -- LAPACK computational routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      CHARACTER          transr, uplo
      INTEGER            info, n
*     .. Array Arguments ..
      REAL               a( 0: * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               one
      parameter                ( one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            lower, nisodd, normaltransr
      INTEGER            n1, n2, k
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, stftri, slauum, strmm, ssyrk
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          mod
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      normaltransr = lsame( transr, 'N' )
      lower = lsame( uplo, 'L' )
      IF( .NOT.normaltransr .AND. .NOT.lsame( transr, 'T' ) ) THEN
         info = -1
      ELSE IF( .NOT.lower .AND. .NOT.lsame( uplo, 'U' ) ) THEN
         info = -2
      ELSE IF( n.LT.0 ) THEN
         info = -3
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPFTRI', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*     Invert the triangular Cholesky factor U or L.
*
      CALL stftri( transr, uplo, 'N', n, a, info )
      IF( info.GT.0 )
     $   RETURN
*
*     If N is odd, set NISODD = .TRUE.
*     If N is even, set K = N/2 and NISODD = .FALSE.
*
      IF( mod( n, 2 ).EQ.0 ) THEN
         k = n / 2
         nisodd = .false.
      ELSE
         nisodd = .true.
      END IF
*
*     Set N1 and N2 depending on LOWER
*
      IF( lower ) THEN
         n2 = n / 2
         n1 = n - n2
      ELSE
         n1 = n / 2
         n2 = n - n1
      END IF
*
*     Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
*     inv(L)^C*inv(L). There are eight cases.
*
      IF( nisodd ) THEN
*
*        N is odd
*
         IF( normaltransr ) THEN
*
*           N is odd and TRANSR = 'N'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
*              T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
*              T1 -> a(0), T2 -> a(n), S -> a(N1)
*
               CALL slauum( 'L', n1, a( 0 ), n, info )
               CALL ssyrk( 'L', 'T', n1, n2, one, a( n1 ), n, one,
     $                     a( 0 ), n )
               CALL strmm( 'L', 'U', 'N', 'N', n2, n1, one, a( n ), n,
     $                     a( n1 ), n )
               CALL slauum( 'U', n2, a( n ), n, info )
*
            ELSE
*
*              SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
*              T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
*              T1 -> a(N2), T2 -> a(N1), S -> a(0)
*
               CALL slauum( 'L', n1, a( n2 ), n, info )
               CALL ssyrk( 'L', 'N', n1, n2, one, a( 0 ), n, one,
     $                     a( n2 ), n )
               CALL strmm( 'R', 'U', 'T', 'N', n1, n2, one, a( n1 ), n,
     $                     a( 0 ), n )
               CALL slauum( 'U', n2, a( n1 ), n, info )
*
            END IF
*
         ELSE
*
*           N is odd and TRANSR = 'T'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is odd
*              T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
*
               CALL slauum( 'U', n1, a( 0 ), n1, info )
               CALL ssyrk( 'U', 'N', n1, n2, one, a( n1*n1 ), n1, one,
     $                     a( 0 ), n1 )
               CALL strmm( 'R', 'L', 'N', 'N', n1, n2, one, a( 1 ), n1,
     $                     a( n1*n1 ), n1 )
               CALL slauum( 'L', n2, a( 1 ), n1, info )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is odd
*              T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
*
               CALL slauum( 'U', n1, a( n2*n2 ), n2, info )
               CALL ssyrk( 'U', 'T', n1, n2, one, a( 0 ), n2, one,
     $                     a( n2*n2 ), n2 )
               CALL strmm( 'L', 'L', 'T', 'N', n2, n1, one, a( n1*n2 ),
     $                     n2, a( 0 ), n2 )
               CALL slauum( 'L', n2, a( n1*n2 ), n2, info )
*
            END IF
*
         END IF
*
      ELSE
*
*        N is even
*
         IF( normaltransr ) THEN
*
*           N is even and TRANSR = 'N'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
*              T1 -> a(1), T2 -> a(0), S -> a(k+1)
*
               CALL slauum( 'L', k, a( 1 ), n+1, info )
               CALL ssyrk( 'L', 'T', k, k, one, a( k+1 ), n+1, one,
     $                     a( 1 ), n+1 )
               CALL strmm( 'L', 'U', 'N', 'N', k, k, one, a( 0 ), n+1,
     $                     a( k+1 ), n+1 )
               CALL slauum( 'U', k, a( 0 ), n+1, info )
*
            ELSE
*
*              SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
*              T1 -> a(k+1,0) ,  T2 -> a(k,0),   S -> a(0,0)
*              T1 -> a(k+1), T2 -> a(k), S -> a(0)
*
               CALL slauum( 'L', k, a( k+1 ), n+1, info )
               CALL ssyrk( 'L', 'N', k, k, one, a( 0 ), n+1, one,
     $                     a( k+1 ), n+1 )
               CALL strmm( 'R', 'U', 'T', 'N', k, k, one, a( k ), n+1,
     $                     a( 0 ), n+1 )
               CALL slauum( 'U', k, a( k ), n+1, info )
*
            END IF
*
         ELSE
*
*           N is even and TRANSR = 'T'
*
            IF( lower ) THEN
*
*              SRPA for LOWER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
*              T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
*
               CALL slauum( 'U', k, a( k ), k, info )
               CALL ssyrk( 'U', 'N', k, k, one, a( k*( k+1 ) ), k, one,
     $                     a( k ), k )
               CALL strmm( 'R', 'L', 'N', 'N', k, k, one, a( 0 ), k,
     $                     a( k*( k+1 ) ), k )
               CALL slauum( 'L', k, a( 0 ), k, info )
*
            ELSE
*
*              SRPA for UPPER, TRANSPOSE, and N is even (see paper)
*              T1 -> B(0,k+1),     T2 -> B(0,k),   S -> B(0,0),
*              T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
*
               CALL slauum( 'U', k, a( k*( k+1 ) ), k, info )
               CALL ssyrk( 'U', 'T', k, k, one, a( 0 ), k, one,
     $                     a( k*( k+1 ) ), k )
               CALL strmm( 'L', 'L', 'T', 'N', k, k, one, a( k*k ), k,
     $                     a( 0 ), k )
               CALL slauum( 'L', k, a( k*k ), k, info )
*
            END IF
*
         END IF
*
      END IF
*
      RETURN
*
*     End of SPFTRI
*

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