# library.linsys Submodule¶

## Module Summary¶

Interfaces for the NAG Mark 28.5 linsys Chapter.

linsys - Simultaneous Linear Equations

This module is concerned with the solution of the matrix equation , where may be a single vector or a matrix of multiple right-hand sides. The matrix may be real, complex, symmetric, Hermitian, positive definite, positive definite Toeplitz or banded. It may also be rectangular, in which case a least squares solution is obtained.

Much of the functionality of this module has been superseded by functions from submodule lapacklin and submodule lapackeig (LAPACK routines) as those modules have grown and have included driver and expert driver functions.

For a general introduction to sparse systems of equations, see the F11 Introduction, which provides functions for large sparse systems. Some functions for sparse problems are also included in this module; they are described in Sparse Matrix Functions.

## Functionality Index¶

Black Box functions,

complex general band matrix: complex_band_solve()

complex general matrix: complex_square_solve()

complex general tridiagonal matrix: complex_tridiag_solve()

complex Hermitian matrix

complex Hermitian positive definite band matrix: complex_posdef_band_solve()

complex Hermitian positive definite matrix

complex Hermitian positive definite tridiagonal matrix: complex_posdef_tridiag_solve()

complex symmetric matrix

real general band matrix: real_band_solve()

real general matrix

multiple right-hand sides, standard precision: real_square_solve()

real general tridiagonal matrix: real_tridiag_solve()

real symmetric matrix

packed matrix format: real_symm_packed_solve()

standard matrix format: real_symm_solve()

real symmetric positive definite band matrix: real_posdef_band_solve()

real symmetric positive definite matrix

multiple right-hand sides, standard precision: real_posdef_solve()

packed matrix format: real_posdef_packed_solve()

real symmetric positive definite Toeplitz matrix

general right-hand side: real_toeplitz_solve()

Yule–Walker equations: real_toeplitz_yule()

real symmetric positive definite tridiagonal matrix: real_posdef_tridiag_solve()

General Purpose functions,

real almost block-diagonal matrix: real_blkdiag_fac_solve()

real band symmetric positive definite matrix, variable bandwidth: real_posdef_vband_solve()

real sparse matrix

real symmetric positive definite Toeplitz matrix

general right-hand side, update solution: real_toeplitz_update()

Yule–Walker equations, update solution: real_toeplitz_yule_update()

real tridiagonal matrix: real_tridiag_fac_solve()

Least squares and Homogeneous Equations

real matrix

, rank or minimal solution: real_gen_solve()

rank , iterative refinement: real_gen_lsqsol()

real sparse matrix: real_gen_sparse_lsqsol()

Service Functions

complex rectangular matrix

norm and condition number estimation: complex_gen_norm_rcomm()

real matrix

covariance matrix for linear least squares problems: real_gen_lsq_covmat()

real rectangular matrix

norm and condition number estimation: real_gen_norm_rcomm()

For full information please refer to the NAG Library document

https://www.nag.com/numeric/nl/nagdoc_28.5/flhtml/f04/f04intro.html