ARTS 2.5.4 (git: 31ce4f0e)
lin_alg.cc File Reference

Linear algebra functions. More...

#include "lin_alg.h"
#include <Eigen/Dense>
#include <Eigen/Eigenvalues>
#include <cmath>
#include <new>
#include <stdexcept>
#include "array.h"
#include "arts.h"
#include "arts_omp.h"
#include "lapack.h"
#include "logic.h"
#include "matpackIII.h"

Go to the source code of this file.

Functions

void ludcmp (Matrix &LU, ArrayOfIndex &indx, ConstMatrixView A)
 LU decomposition. More...
 
void lubacksub (VectorView x, ConstMatrixView LU, ConstVectorView b, const ArrayOfIndex &indx)
 LU backsubstitution. More...
 
void solve (VectorView x, ConstMatrixView A, ConstVectorView b)
 Solve a linear system. More...
 
void inv (MatrixView Ainv, ConstMatrixView A)
 Matrix Inverse. More...
 
void inv (ComplexMatrixView Ainv, const ConstComplexMatrixView A)
 
void diagonalize (MatrixView P, VectorView WR, VectorView WI, ConstMatrixView A)
 Matrix Diagonalization. More...
 
void diagonalize (ComplexMatrixView P, ComplexVectorView W, const ConstComplexMatrixView A)
 Matrix Diagonalization. More...
 
void matrix_exp (MatrixView F, ConstMatrixView A, const Index &q)
 General exponential of a Matrix. More...
 
Numeric norm_inf (ConstMatrixView A)
 Maximum absolute row sum norm. More...
 
void id_mat (MatrixView I)
 Identity Matrix. More...
 
Numeric det (ConstMatrixView A)
 
void linreg (Vector &p, ConstVectorView x, ConstVectorView y)
 
Numeric lsf (VectorView x, ConstMatrixView A, ConstVectorView y, bool residual) noexcept
 Least squares fitting by solving x for known A and y. More...
 

Detailed Description

Linear algebra functions.

Author
Claudia Emde claud.nosp@m.ia.e.nosp@m.mde@d.nosp@m.lr.d.nosp@m.e
Date
Thu May 2 10:59:55 2002

This file contains mathematical tools to solve the vector radiative transfer equation.

Definition in file lin_alg.cc.

Function Documentation

◆ det()

Determinant of N by N matrix. Simple recursive method.

Parameters
AIn: Matrix of size NxN.
Author
Richard Larsson
Date
2012-08-03

Definition at line 504 of file lin_alg.cc.

References ARTS_ASSERT, det(), ConstMatrixView::ncols(), ConstMatrixView::nrows(), and temp.

Referenced by det().

◆ diagonalize() [1/2]

void diagonalize ( ComplexMatrixView  P,
ComplexVectorView  W,
const ConstComplexMatrixView  A 
)

Matrix Diagonalization.

Return P and W from A in the statement diag(P^-1*A*P)-W == 0.

The function makes many copies and is thereby not fast. There are no tests on the condition of the returned matrix, so nan and inf can occur.

Parameters
[out]PThe right eigenvectors.
[out]WThe eigenvalues.
[in]AThe matrix to diagonalize.

Definition at line 335 of file lin_alg.cc.

References ARTS_ASSERT, ConstComplexVectorView::get_c_array(), ConstComplexMatrixView::get_c_array(), ConstVectorView::get_c_array(), ConstComplexMatrixView::get_column_extent(), ConstComplexMatrixView::ncols(), ConstComplexVectorView::nelem(), ConstComplexMatrixView::nrows(), swap(), and lapack::zgeev_().

◆ diagonalize() [2/2]

void diagonalize ( MatrixView  P,
VectorView  WR,
VectorView  WI,
ConstMatrixView  A 
)

Matrix Diagonalization.

Return P and W from A in the statement diag(P^-1*A*P)-W == 0. The real function will require some manipulation if the eigenvalues are imaginary.

The real version returns WR and WI as returned by dgeev. The complex version just returns W.

The function makes many copies and is thereby not fast. There are no tests on the condition of the returned matrix, so nan and inf can occur.

Parameters
[out]PThe right eigenvectors.
[out]WRThe real eigenvalues.
[out]WIThe imaginary eigenvalues.
[in]AThe matrix to diagonalize.

Definition at line 250 of file lin_alg.cc.

Referenced by Absorption::LineMixing::EquivalentLines::EquivalentLines(), lm_hitran_2017::eqvlines(), test_complex_diagonalize(), and test_real_diagonalize().

◆ id_mat()

◆ inv() [1/2]

◆ inv() [2/2]

void inv ( MatrixView  Ainv,
ConstMatrixView  A 
)

Matrix Inverse.

Compute the inverse of a matrix such that I = Ainv*A = A*Ainv. Both MatrixViews must be square and have the same size n. During the inversion one additional n times n Matrix is allocated and work space memory for faster inversion is allocated and freed.

Parameters
[out]AinvThe MatrixView to contain the inverse of A.
[in]AThe matrix to be inverted.

Definition at line 168 of file lin_alg.cc.

Referenced by benchmark_inv(), benchmark_oem_linear(), lm_hitran_2017::eqvlines(), LineShape::ComputeData::interp_add_even(), LineShape::ComputeData::interp_add_triplequad(), CovarianceMatrix::invert_correlation_block(), test_addition(), test_complex_diagonalize(), test_inv(), test_oem_gauss_newton(), test_oem_levenberg_marquardt(), and test_real_diagonalize().

◆ linreg()

void linreg ( Vector p,
ConstVectorView  x,
ConstVectorView  y 
)

Determines coefficients for linear regression

Performs a least squares estimation of the model

y = p[0] + p[1] * x

Parameters
pOut: Fitted coefficients.
xIn: x-value of data points
yIn: y-value of data points
Author
Patrick Eriksson
Date
2013-01-25

Definition at line 548 of file lin_alg.cc.

References ARTS_ASSERT, ConstVectorView::nelem(), and Vector::resize().

Referenced by derive_scat_species_a_and_b(), and ppathFromRtePos2().

◆ lsf()

Numeric lsf ( VectorView  x,
ConstMatrixView  A,
ConstVectorView  y,
bool  residual = true 
)
noexcept

Least squares fitting by solving x for known A and y.

(A^T A)x = A^T y

Returns the squared residual, i.e., <(A^T A)x-A^T y|(A^T A)x-A^T y>.

Parameters
[in]xAs equation
[in]AAs equation
[in]yAs equation
[in]residual(optional) Returns the residual if true
Returns
Squared residual or 0

Definition at line 596 of file lin_alg.cc.

◆ lubacksub()

void lubacksub ( VectorView  x,
ConstMatrixView  LU,
ConstVectorView  b,
const ArrayOfIndex indx 
)

LU backsubstitution.

Solves a set of linear equations Ax=b. It is neccessairy to do a L decomposition using the function ludcp before using this function. The backsubstitution is in-place, i.e. x and b may be the same vector.

Parameters
xOutput: Solution vector of the equation system.
LUInput: LU decomposition of the matrix (output of function ludcp).
bInput: Right-hand-side vector of equation system.
indxInput: Pivoting information (output of function ludcp).

Definition at line 92 of file lin_alg.cc.

Referenced by matrix_exp(), solve(), test_lusolve1D(), test_lusolve4D(), and test_solve_linear_system().

◆ ludcmp()

void ludcmp ( Matrix LU,
ArrayOfIndex indx,
ConstMatrixView  A 
)

LU decomposition.

This function performes a LU Decomposition of the matrix A. (Compare Numerical Recipies in C, pages 36-48.)

Parameters
LUOutput: returns L and U in one matrix
indxOutput: Vector that records the row permutation.
AInput: Matrix for which the LU decomposition is performed

Definition at line 57 of file lin_alg.cc.

Referenced by matrix_exp(), solve(), test_lusolve1D(), test_lusolve4D(), and test_solve_linear_system().

◆ matrix_exp()

void matrix_exp ( MatrixView  F,
ConstMatrixView  A,
const Index q 
)

General exponential of a Matrix.

The exponential of a matrix is computed using the Pade-Approximation. The method is decribed in: Golub, G. H. and C. F. Van Loan, Matrix Computation, p. 384, Johns Hopkins University Press, 1983.

The Pade-approximation is applied on all cases. If a faster option can be applied has to be checked before calling the function.

Parameters
FOutput: The matrix exponential of A (Has to be initialized before calling the function.
AInput: arbitrary square matrix
qInput: Parameter for the accuracy of the computation

Definition at line 396 of file lin_alg.cc.

References ARTS_ASSERT, c, id_mat(), is_size(), joker, Constant::k, lubacksub(), ludcmp(), mult(), N, ConstMatrixView::ncols(), norm_inf(), pow(), and q.

Referenced by ext2trans(), test_matrix_exp1D(), test_matrix_exp3D(), test_matrix_exp4D(), and test_real_diagonalize().

◆ norm_inf()

Numeric norm_inf ( ConstMatrixView  A)

Maximum absolute row sum norm.

This function returns the maximum absolute row sum norm of a matrix A (see user guide for the definition).

Parameters
AInput: arbitrary matrix
Returns
Maximum absolute row sum norm

Definition at line 471 of file lin_alg.cc.

References abs, ConstMatrixView::ncols(), norm_inf(), and ConstMatrixView::nrows().

Referenced by matrix_exp(), and norm_inf().

◆ solve()

void solve ( VectorView  x,
ConstMatrixView  A,
ConstVectorView  b 
)

Solve a linear system.

Solves the linear system A*x = b for a general matrix A. For the solution of the system an additional n-times-n matrix and a size-n index vector are allocated.

Parameters
xThe solution vector x.
AThe matrix A defining the system.
bThe vector b.

Definition at line 139 of file lin_alg.cc.

References ARTS_ASSERT, b, lubacksub(), ludcmp(), ConstMatrixView::ncols(), ConstVectorView::nelem(), and ConstMatrixView::nrows().