linalg::solve_triangular_system Interface Reference

Solves a triangular system of equations. More...

Public Member Functions
	solve_tri_mtx
	solve_tri_mtx_cmplx
	solve_tri_vec
	solve_tri_vec_cmplx

Detailed Description

Solves a triangular system of equations.

Syntax 1

Solves one of the matrix equations: \( op(A) X = \alpha B \), or \( X op(A) = \alpha B \), where \( A \) is a triangular matrix.

subroutine solve_triangular_system(logical lside, logical upper, logical trans, logical nounit, real(real64) alpha, real(real64) a(:,:), real(real64) b(:,:), optional class(errors) err)
subroutine solve_triangular_system(logical lside, logical upper, logical trans, logical nounit, complex(real64) alpha, complex(real64) a(:,:), complex(real64) b(:,:), optional class(errors) err)

Parameters

[in]	lside	Set to true to solve \( op(A) X = \alpha B \); else, set to false to solve \( X op(A) = \alpha B \).
[in]	upper	Set to true if A is an upper triangular matrix; else, set to false if A is a lower triangular matrix.
[in]	trans	Set to true if \( op(A) = A^T \) ( \( op(A) = A^H \) in the complex case); else, set to false if \( op(A) = A \).
[in]	nounit	Set to true if A is not a unit-diagonal matrix (ones on every diagonal element); else, set to false if A is a unit-diagonal matrix.
[in]	alpha	The scalar multiplier to B.
[in]	a	If lside is true, the M-by-M triangular matrix on which to operate; else, if lside is false, the N-by-N triangular matrix on which to operate.
[in,out]	b	On input, the M-by-N right-hand-side. On output, the M-by-N solution.
[in,out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. LA_ARRAY_SIZE_ERROR: Occurs if a is not square, or if the sizes of a and b are not compatible.

Notes

This routine is based upon the BLAS routine DTRSM (ZTRSM in the complex case).

Syntax 2

Solves the system of equations: \( op(A) X = B \), where \( A \) is a triangular matrix.

subroutine solve_triangular_system(logical upper, logical trans, logical nounit, real(real64) a(:,:), real(real64) x(:), optional class(errors) err)
subroutine solve_triangular_system(logical upper, logical trans, logical nounit, complex(real64) a(:,:), complex(real64) x(:), optional class(errors) err)

Parameters

[in]	upper	Set to true if A is an upper triangular matrix; else, set to false if A is a lower triangular matrix.
[in]	trans	Set to true if \( op(A) = A^T \) ( \( op(A) = A^H \) in the complex case); else, set to false if \( op(A) = A \).
[in]	nounit	Set to true if A is not a unit-diagonal matrix (ones on every diagonal element); else, set to false if A is a unit-diagonal matrix.
[in]	a	The N-by-N triangular matrix.
[in,out]	x	On input, the N-element right-hand-side array. On output, the N-element solution array.
[in,out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. LA_ARRAY_SIZE_ERROR: Occurs if a is not square, or if the sizes of a and b are not compatible.

Notes

This routine is based upon the BLAS routine DTRSV (ZTRSV in the complex case).

Usage

The following example illustrates the solution of two triangular systems to solve a system of LU factored equations.

program example
    use iso_fortran_env, only : real64, int32
    use linalg
    implicit none
 
    ! Variables
    real(real64) :: a(3,3), b(3), u(3,3), p(3,3)
    integer(int32) :: i, pvt(3)
 
    ! Build the 3-by-3 matrix A.
    !     | 1   2   3 |
    ! A = | 4   5   6 |
    !     | 7   8   0 |
    a = reshape( &
        [1.0d0, 4.0d0, 7.0d0, 2.0d0, 5.0d0, 8.0d0, 3.0d0, 6.0d0, 0.0d0], &
        [3, 3])
 
    ! Build the right-hand-side vector B.
    !     | -1 |
    ! b = | -2 |
    !     | -3 |
    b = [-1.0d0, -2.0d0, -3.0d0]
 
    ! The solution is:
    !     |  1/3 |
    ! x = | -2/3 |
    !     |   0  |
 
    ! Compute the LU factorization
    call lu_factor(a, pvt)
 
    ! Extract the L and U matrices. A is overwritten with L.
    call form_lu(a, pvt, u, p)
 
    ! Solve the lower triangular system L * Y = P * B for Y, but first compute
    ! P * B, and store the results in B
    b = matmul(p, b)
 
    ! Now, compute the solution to the lower triangular system.  Store the
    ! result in B.  Remember, L is unit diagonal (ones on its diagonal)
    call solve_triangular_system(.false., .false., .false., a, b)
 
    ! Solve the upper triangular system U * X = Y for X.
    call solve_triangular_system(.true., .false., .true., u, b)
 
    ! Display the results.
    print '(A)', "LU Solution: X = "
    print '(F8.4)', (b(i), i = 1, size(b))
end program

The above program produces the following output.

LU Solution: X =
0.3333
-0.6667
0.0000

Definition at line 2316 of file linalg.f90.

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The documentation for this interface was generated from the following file:

• D:/Code/linalg/src/linalg.f90