Computes transpose matrix-matrix or transpose matrix-vector product.

Operator Return Value

Matrix containing the product of AT and B.   (Output)

Required Operands

A — Left operand matrix. This is an array of rank 2, or 3. It may be real, double, complex, double complex, or one of the computational sparse matrix derived types, ?_hbc_sparse. (Input)
Note that A and B cannot both be ?_hbc_sparse.

B — Right operand matrix or vector. This is an array of rank 1, 2, or 3. It may be real, double, complex, double complex, or one of the computational sparse matrix derived types, ?_hbc_sparse. (Input)
Note that A and B cannot both be ?_hbc_sparse.
 
If A has rank three, B must have rank three.
If B has rank three, A must have rank three.

FORTRAN 90 Interface

A .tx. B

Description

Computes the product of the transpose of  matrix A and matrix or vector B. The results are in a precision and data type that ascends to the most accurate or complex operand.

Rank three operation is defined as follows:

do i = 1, min(size(A,3), size(B,3))
              X(:,:,i) =  A(:,:,i) .tx. B(:,:,i)
       end do

 

.tx. can be used with either dense or sparse matrices. It is MPI capable for dense matrices only.

Examples

Dense Matrix Example (operator_ex05.f90)

 

use linear_operators

      implicit none

 

! This is the equivalent of Example 1 for LIN_SOL_SELF using operators
! and functions.

      integer, parameter :: m=64, n=32

      real(kind(1e0)) :: one=1.0e0, err

      real(kind(1e0)) A(n,n), b(n,n), C(m,n), d(m,n), x(n,n)

 

! Generate two rectangular random matrices.

      C = rand(C); d=rand(d)

 

! Form the normal equations for the rectangular system.

      A = C .tx. C; b = C .tx. d

 

! Compute the solution for Ax = b, A is symmetric.

      x = A .ix. b

 

! Check the results.

      err = norm(b - (A .x. x))/(norm(A)+norm(b))

      if (err <= sqrt(epsilon(one))) then

         write (*,*) 'Example 1 for LIN_SOL_SELF (operators) is correct.'

      end if

 

      end 

Sparse Matrix Example

 

 use wrrrn_int

 use linear_operators

 

 type (s_sparse) S

 type (s_hbc_sparse) H

 integer, parameter :: N=3

 real (kind(1.e0)) x(N,N), y(N,N), B(N,N)

 real (kind(1.e0)) err

 

 S = s_entry (1, 1, 2.0)

 S = s_entry (1, 3, 1.0)

 S = s_entry (2, 2, 4.0)

 S = s_entry (3, 3, 6.0)

 H = S   ! sparse

 X = H   ! dense equivalent of H

 B = rand(B)

 Y = H .tx. B

  call wrrrn ( 'H', X)

  call wrrrn ( 'B', b)

  call wrrrn ( 'H .tx. B ', y)

 

! Check the results.

     err =  norm(y - (X .tx. B))

      if (err <= sqrt(epsilon(one))) then

         write (*,*) 'Sparse example for .tx. operator is correct.'

      end if

 

 end

    Output

 

             H

         1       2       3

 1   2.000   0.000   1.000

 2   0.000   4.000   0.000

 3   0.000   0.000   6.000

 

               B

          1        2        3

 1   0.8711   0.4467   0.4743

 2   0.8315   0.7257   0.4518

 3   0.6839   0.0561   0.6972

 

         H .tx. B

         1       2       3

 1   1.742   0.893   0.949

 2   3.326   2.903   1.807

 3   4.975   0.784   4.657

 Sparse example for .tx. operator is correct.

Parallel Example  (parallel_ex05.f90)

 

      use linear_operators

      use mpi_setup_int

 

      implicit none

 

! This is the equivalent of Parallel Example 5 using box data types,
! operators and functions.

 

      integer, parameter :: m=64, n=32, nr=4

      real(kind(1e0)) :: one=1e0, err(nr)

      real(kind(1e0)), dimension(n,n,nr) :: A, b, x

      real(kind(1e0)), dimension(m,n,nr) :: C, d

 

! Setup for MPI.

      mp_nprocs = mp_setup()

 

! Generate two rectangular random matrices, only

! at the root node.

      if (mp_rank == 0) then

      C = rand(C); d=rand(d)

      endif

 

! Form the normal equations for the rectangular system.

      A = C .tx. C; b = C .tx. d

 

! Compute the solution for Ax = b.

      x = A .ix. b

 

! Check the results.

      err = norm(b - (A .x. x))/(norm(A)+norm(b))

      if (ALL(err <= sqrt(epsilon(one))) .AND. MP_RANK == 0) &

         write (*,*) 'Parallel Example 5 is correct.'

     

! See to any error messages and quit MPI.

      mp_nprocs = mp_setup('Final')

 

      end

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