.hx.

Chapter 10: Linear Algebra Operators and Generic Functions

.hx.

Computes conjugate transpose matrix-matrix product.

Operator Return Value

Matrix containing the product of A H 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 .hx. B

Description

Computes the product of the conjugate 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) .hx. B(:,:,i)
      end do

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

Examples

Dense Matrix Example (operator_ex32.f90)

use linear_operators

implicit none

! This is the equivalent of Example 4 (using operators) for LIN_EIG_GEN.

integer, parameter :: n=17

real(kind(1d0)), parameter :: one=1d0

real(kind(1d0)), dimension(n,n) :: A, C

real(kind(1d0)) variation(n), eta

complex(kind(1d0)), dimension(n,n) :: U, V, e(n), d(n)

! Generate a random matrix.

A = rand(A)

! Compute the eigenvalues, left- and right- eigenvectors.

D = EIG(A, W=V); E = EIG(.t.A, W=U)

! Compute condition numbers and variations of eigenvalues.

variation = norm(A)/abs(diagonals( U .hx. V))

! Now perturb the data in the matrix by the relative factors

! eta=sqrt(epsilon) and solve for values again. Check the

! differences compared to the estimates. They should not exceed

! the bounds.

eta = sqrt(epsilon(one))

C = A + eta*(2*rand(A)-1)*A

D = EIG(C)

! Looking at the differences of absolute values accounts for

! switching signs on the imaginary parts.

if (count(abs(d)-abs(e) > eta*variation) == 0) then

write (*,*) 'Example 4 for LIN_EIG_GEN (operators) is correct.'

end if

end

Sparse Matrix Example

use wrcrn_int

use linear_operators

type (c_sparse) S

type (c_hbc_sparse) H

integer, parameter :: N=3

complex (kind(1.e0)) x(N,N), y(N,N), A(N,N)

real (kind(1.e0)) err

S = c_entry (1, 1, (2.0, 1.0) )

S = c_entry (1, 3, (1.0, 3.0))

S = c_entry (2, 2, (4.0, -1.0))

S = c_entry (3, 3, (6.0, 2.0))

H = S ! sparse

X = H ! dense equivalent of H

A= rand(A)

Y = H .hx. A

call wrcrn ( 'H', X)

call wrcrn ( 'A', a)

call wrcrn ( 'H .hx. A ', y)

! Check the results.

err = norm(y - (X .hx. A))

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

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

end if

end

Output

1 2 3

1 ( 2.000, 1.000) ( 0.000, 0.000) ( 1.000, 3.000)

2 ( 0.000, 0.000) ( 4.000,-1.000) ( 0.000, 0.000)

3 ( 0.000, 0.000) ( 0.000, 0.000) ( 6.000, 2.000)

1 2 3

1 ( 0.6278, 0.8475) ( 0.8007, 0.4179) ( 0.4512, 0.2601)

2 ( 0.1249, 0.4675) ( 0.7957, 0.1609) ( 0.4228, 0.0507)

3 ( 0.4608, 0.0891) ( 0.3181, 0.9180) ( 0.9961, 0.1939)

H .hx. A

1 2 3

1 ( 2.103, 1.067) ( 2.019, 0.035) ( 1.163, 0.069)

2 ( 0.032, 1.995) ( 3.022, 1.439) ( 1.640, 0.626)

3 ( 6.113,-1.423) ( 5.799, 2.888) ( 7.596,-1.922)

Sparse example for .hx. operator is correct.

Parallel Example

use linear_operators

use mpi_setup_int

integer, parameter :: N=32, nr=4

complex (kind(1.e0)) A(N,N,nr), B(N,N,nr), Y(N,N,nr)

! Setup for MPI

mp_nprocs = mp_setup()

if (mp_rank == 0) then

A = rand(A)

B = rand(B)

end if

Y = A .hx. B

mp_nprocs = mp_setup ('Final')

end

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