Solves a system of M QR-factored equations of N unknowns. More...

Detailed Description

Solves a system of M QR-factored equations of N unknowns.

Syntax 1 (No Pivoting): subroutine solve_qr(real(real64) a(:,:), real(real64) tau(:), real(real64) b(:,:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(complex(real64) a(:,:), complex(real64) tau(:), complex(real64) b(:,:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(real(real64) a(:,:), real(real64) tau(:), real(real64) b(:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(complex(real64) a(:,:), complex(real64) tau(:), complex(real64) b(:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in]	a	On input, the M-by-N QR factored matrix as returned by qr_factor. On output, the contents of this matrix are restored. Notice, M must be greater than or equal to N.
[in]	tau	A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.
[in]	b	On input, the M-by-NRHS right-hand-side matrix. On output, the first N rows are overwritten by the solution matrix X.
[out]	work	An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least `olwork`.
[out]	olwork	An optional output used to determine workspace size. If supplied, the routine determines the optimal size for `work`, and returns without performing any actual calculations.
[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 any of the input arrays are not sized appropriately. LA_OUT_OF_MEMORY_ERROR: Occurs if local memory must be allocated, and there is insufficient memory available.

Syntax 2 (With Pivoting): subroutine solve_qr(real(real64) a(:,:), real(real64) tau(:), integer(int32) jpvt(:), real(real64) b(:,:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(complex(real64) a(:,:), complex(real64) tau(:), integer(int32) jpvt(:), complex(real64) b(:,:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(real(real64) a(:,:), real(real64) tau(:), integer(int32) jpvt(:), real(real64) b(:), optional real(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

subroutine solve_qr(complex(real64) a(:,:), complex(real64) tau(:), integer(int32) jpvt(:), complex(real64) b(:), optional complex(real64) work(:), optional integer(int32) olwork, optional class(errors) err)

Parameters

[in]	a	On input, the M-by-N QR factored matrix as returned by qr_factor. On output, the contents of this matrix are altered.
[in]	tau	A MIN(M, N)-element array containing the scalar factors of the elementary reflectors as returned by qr_factor.
[in]	jpvt	An N-element array, as output by qr_factor, used to track the column pivots.
[in]	b	On input, the MAX(M, N)-by-NRHS matrix where the first M rows contain the right-hand-side matrix B. On output, the first N rows are overwritten by the solution matrix X.
[out]	work	An optional input, that if provided, prevents any local memory allocation. If not provided, the memory required is allocated within. If provided, the length of the array must be at least `olwork`.
[out]	olwork	An optional output used to determine workspace size. If supplied, the routine determines the optimal size for `work`, and returns without performing any actual calculations.
[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 any of the input arrays are not sized appropriately. LA_OUT_OF_MEMORY_ERROR: Occurs if local memory must be allocated, and there is insufficient memory available.

Usage: The following example illustrates the solution of a system of equations using QR factorization.
program example

use iso_fortran_env, only : real64, int32

use linalg

implicit none

! Local Variables

real(real64) :: a(3,3), tau(3), b(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 QR factorization, using pivoting

pvt = 0 ! Zero every entry in order not to lock any column in place

call qr_factor(a, tau, pvt)

! Compute the solution. The results overwrite b.

call solve_qr(a, tau, pvt, b)

! Display the results.

print '(A)', "QR Solution: X = "

print '(F8.4)', (b(i), i = 1, size(b))

! Notice, QR factorization without pivoting could be accomplished in the

! same manner. The only difference is to omit the PVT array (column pivot

! tracking array).

end program

linalg::qr_factor
Computes the QR factorization of an M-by-N matrix.
Definition linalg.f90:1121

linalg::solve_qr
Solves a system of M QR-factored equations of N unknowns.
Definition linalg.f90:2557

linalg
Provides a set of common linear algebra routines.
Definition linalg.f90:145

The above program produces the following output.
QR Solution: X =

0.3333

-0.6667

0.0000

See Also

Definition at line 2557 of file linalg.f90.

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

Public Member Functions