# Linear Systems

## LINEAR SOLVERS

ROUTINE | DESCRIPTION |
---|---|

LIN_SOL_GEN | Solves a real general system of linear equations Ax = b. |

LIN_SOL_SELF | Solves a system of linear equations Ax = b, where A is a self-adjoint matrix. |

LIN_SOL_LSQ | Solves a rectangular system of linear equations Ax ≅ b, in a least-squares sense. |

LIN_SOL_SVD | Solves a rectangular least-squares system of linear equations Ax ≅ b using singular value decomposition. |

LIN_SOL_TRI | Solves multiple systems of linear equations. |

LIN_SVD | Computes the singular value decomposition (SVD) of a rectangular matrix, A. |

## LARGE-SCALE PARALLEL SOLVERS

ROUTINE | DESCRIPTION |
---|---|

PARALLEL_NONNEGATIVE_LSQ | Solves a linear, non-negative constrained least-squares system. |

PARALLEL_BOUNDED_LSQ | Solves a linear least-squares system with bounds on the unknowns. |

## SOLUTION OF LINEAR SYSTEMS, MATRIX INVERSION, AND DETERMINANT EVALUATION

### REAL GENERAL MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSARG | Solves a real general system of linear equations with iterative refinement. |

LSLRG | Solves a real general system of linear equations without iterative refinement. |

LFCRG | Computes the LU factorization of a real general matrix and estimates its L condition number._{1} |

LFTRG | Computes the LU factorization of a real general matrix. |

LFSRG | Solves a real general system of linear equations given the LU factorization of the coefficient matrix. |

LFIRG | Uses iterative refinement to improve the solution of a real general system of linear equations. |

LFDRG | Computes the determinant of a real general matrix given the LU factorization of the matrix. |

LINRG | Computes the inverse of a real general matrix. |

### COMPLEX GENERAL MATRICESS

ROUTINE | DESCRIPTION |
---|---|

LSACG | Solves a complex general system of linear equations with iterative refinement. |

LSLCG | Solves a complex general system of linear equations without iterative refinement. |

LFCCG | Computes the LU factorization of a complex general matrix and estimates its L condition number._{1} |

LFTCG | Computes the LU factorization of a complex general matrix. |

LFSCG | Solves a complex general system of linear equations given the LU factorization of the coefficient matrix. |

LFICG | Uses iterative refinement to improve the solution of a complex general system of linear equations. |

LFDCG | Computes the determinant of a complex general matrix given the LU factorization of the matrix. |

LINCG | Computes the inverse of a complex general matrix. |

### REAL TRIANGULAR MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSLRT | Solves a real triangular system of linear equations. |

LFCRT | Estimates the condition number of a real triangular matrix. |

LFDRT | Computes the determinant of a real triangular matrix. |

LINRT | Computes the inverse of a real triangular matrix. |

### COMPLEX TRIANGULAR MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSLCT | Solves a complex triangular system of linear equations. |

LFCCT | Estimates the condition number of a complex triangular matrix. |

LFDCT | Computes the determinant of a complex triangular matrix. |

LINCT | Computes the inverse of a complex triangular matrix. |

### REAL POSITIVE DEFINITE MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSADS | Solves a real symmetric positive definite system of linear equations with iterative refinement. |

LSLDS | Solves a real symmetric positive definite system of linear equations without iterative refinement. |

LFCDS | Computes the R Cholesky factorization of a real symmetric positive definite matrix and estimates its^{T}R Lcondition number._{1} |

LFTDS | Computes the RCholesky factorization of a real symmetric positive definite matrix.^{T}R |

LFSDS | Solves a real symmetric positive definite system of linear equations given the R Cholesky factorization of the coefficient matrix.^{T}R |

LFIDS | Uses the iterative refinement to improve the solution of a real symmetric positive definite system of linear equations. |

LFDDS | Computes the determinant of a real symmetric positive matrix given the R Cholesky factorization of the matrix.^{T}R |

LINDS | Computes the inverse of a real symmetric positive definite matrix. |

### REAL SYMMETRIC MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSASF | Solves a real symmetric system of linear equations with iterative refinement. |

LSLSF | Solves a real symmetric system of linear equations without iterative refinement. |

LFCSF | Computes the U DU factorization of a real symmetric matrix and estimates its^{T} L condition number._{1} |

LFTSF | Computes the U DU factorization of a real symmetric matrix.^{T} |

LFSSF | Solves a real symmetric system of linear equations given the U DU factorization of the coefficient matrix.^{T} |

LFISF | Uses iterative refinement to improve the solution of a real symmetric system of linear equations. |

LFDSF | Computes the determinant of a real symmetric matrix given the U DU factorization of the matrix.^{T} |

### COMPLEX HERMITIAN POSITIVE DEFINITE MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSADH | Solves a complex Hermitian positive definite system of linear equations with iterative refinement. |

LSLDH | Solves a complex Hermitian positive definite system of linear equations without iterative refinement. |

LFCDH | Computes the Rfactorization of a complex Hermitian positive definite matrix and estimates its ^{H}R L condition number._{1} |

LFTDH | Computes the R factorization of a complex Hermitian positive definite matrix.^{H}R |

LFSDH | Solves a complex Hermitian positive definite system of linear equations given the R factorization of the coefficient matrix.^{H}R |

LFIDH | Uses the iterative refinement to improve the solution of a complex Hermitian positive definite system of linear equations. |

LFDDH | Computes the determinant of a complex Hermitian positive definite matrix given the R Cholesky factorization of the matrix.^{H}R |

### COMPLEX HERMITIAN MATRICES

ROUTINE | DESCRIPTION |
---|---|

LSAHF | Solves a complex Hermitian system of linear equations with iterative refinement. |

LSLHF | Solves a complex Hermitian system of linear equations without iterative refinement. |

LFCHF | Computes the U DU factorization of a complex Hermitian matrix and estimates its ^{H}L condition number._{1} |

LFTHF | Computes the U DU factorization of a complex Hermitian matrix.^{H} |

LFSHF | Solves a complex Hermitian system of linear equations given the U DUfactorization of the coefficient matrix.^{H} |

LFIHF | Uses iterative refinement to improve the solution of a complex Hermitian system of linear equations. |

LFDHF | Computes the determinant of a complex Hermitian matrix given the U DU factorization of the matrix.^{H} |

### REAL BAND MATRICES IN BAND STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSLTR | Solves a real tridiagonal system of linear equations. |

LSLCR | Computes the L DU factorization of a a real tridiagonal matrix A using a cyclic reduction algorithm. |

LSARB | Solves a real system of linear equations in band storage mode with iterative refinement. |

LSLRB | Solves a real system of linear equations in band storage mode without iterative refinement. |

LFCRB | Computes the LU factorization of a real matrix in band storage mode and estimates its L condition number._{1} |

LFTRB | Computes the LU factorization of a real matrix in band storage mode. |

LFSRB | Solves a real system of linear equations given the LU factorization of the coefficient matrix in band storage mode. |

LFIRB | Uses iterative refinement to improve the solution of a real system of linear equations in band storage mode. |

LFDRB | Computes the determinant of a real matrix in band storage mode given the LU factorization of the matrix. |

### REAL BAND SYMMETRIC POSITIVE DEFINITE MATRICES IN BAND STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSAQS | Solves a real symmetric positive definite system of linear equations in band symmetric storage mode with iterative refinement. |

LSLQS | Solves a real symmetric positive definite system of linear equations in band symmetric storage mode without iterative refinement. |

LSLPB | Computes the R Cholesky factorization of a real symmetric positive definite matrix ^{T}DRA in codiagonal band symmetric storage mode. Solves a system Ax = b. |

LFCQS | Computes the R Cholesky factorization of a real symmetric positive definite matrix in band symmetric storage mode and estimates its^{T}R L condition number._{1} |

LFTQS | Computes the R Cholesky factorization of a real symmetric positive definite matrix in band symmetric storage mode.^{T}R |

LFSQS | Solves a real symmetric positive definite system of linear equations given the factorization of the coefficient matrix in band symmetric storage mode. |

LFIQS | Uses iterative refinement to improve the solution of a real symmetric positive definite system of linear equations in band symmetric storage mode. |

LFDQS | Computes the determinant of a real symmetric positive definite matrix given the R Cholesky factorization of the matrix in band symmetric storage mode.^{T}R |

### COMPLEX BAND MATRICES IN BAND STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSLTQ | Solves a complex tridiagonal system of linear equations. |

LSLCQ | Computes the LDU factorization of a complex tridiagonal matrix A using a cyclic reduction algorithm. |

LSACB | Solves a complex system of linear equations in band storage mode with iterative refinement. |

LSLCB | Solves a complex system of linear equations in band storage mode without iterative refinement. |

LFCCB | Computes the LU factorization of a complex matrix in band storage mode and estimates its L condition number.the _{1}U DU factorization of the coefficient matrix.^{H} |

LFTCB | Computes the LU factorization of a complex matrix in band storage mode given the coefficient matrix in band storage mode. |

LFSCB | Solves a complex system of linear equations given the LU factorization of the coefficient matrix in band storage mode. |

LFICB | Uses iterative refinement to improve the solution of a complex system of linear equations in band storage mode. |

LFDCB | Computes the determinant of a complex matrix given the LU factorization of the matrix in band storage mode. |

### COMPLEX BAND POSITIVE DEFINITE MATRICES IN BAND STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSAQH | Solves a complex Hermitian positive definite system of linear equations in band Hermitian storage mode with iterative refinement. |

LSLQH | Solves a complex Hermitian positive definite system of linear equations in band Hermitian storage mode without iterative refinement. |

LSLQB | Computes the R Cholesky factorization of a complex Hermitian positive-definite matrix ^{H}DRA in codiagonal band Hermitian storage mode. Solves a system Ax = b. |

LFCQH | Computes the R factorization of a complex Hermitian positive definite matrix in band Hermitian storage mode and estimates its ^{H}RL_{1 }condition number. |

LFTQH | Computes the R factorization of a complex Hermitian positive definite matrix in band Hermitian storage mode.^{H}R |

LFSQH | Solves a complex Hermitian positive definite system of linear equations given the factorization of the coefficient matrix in band Hermitian storage mode. |

LFIQH | Uses iterative refinement to improve the solution of a complex Hermitian positive definite system of linear equations in band Hermitian storage mode. |

LFDQH | Computes the determinant of a complex Hermitian positive definite matrix given the RCholesky factorization in band Hermitian storage mode.^{H}R |

### REAL SPARSE LINEAR EQUATION SOLVERS

ROUTINE | DESCRIPTION |
---|---|

LSLXG | Solves a sparse system of linear algebraic equations by Gaussian elimination. |

LFTXG | Computes the LU factorization of a real general sparse matrix. |

LFSXG | Solves a sparse system of linear equations given the LU factorization of the coefficient matrix. |

### COMPLEX SPARSE LINEAR EQUATION SOLVERS

ROUTINE | DESCRIPTION |
---|---|

LSLZG | Solves a complex sparse system of linear equations by Gaussian elimination. |

LSTZG | Computes the LU factorization of a complex general sparse matrix. |

LFSZG | Solves a complex sparse system of linear equations given the LU factorization of the coefficient matrix. |

### REAL SPARSE SYMMETRIC POSITIVE DEFINITE LINEAR EQUATIONS SOLVERS

ROUTINE | DESCRIPTION |
---|---|

LSLXD | Solves a sparse system of symmetric positive definite linear algebraic equations by Gaussian elimination. |

LSCXD | Performs the symbolic Cholesky factorization for a sparse symmetric matrix using a minimum degree ordering or a user-specified ordering, and sets up the data structure for the numerical Cholesky factorization. |

LNFXD | Computes the numerical Cholesky factorization of a sparse symmetrical matrix A. |

LFSXD | Solves a real sparse symmetric positive definite system of linear equations, given the Cholesky factorization of the coefficient matrix. |

### COMPLEX SPARSE HERMITIAN POSITIVE DEFINITE LINEAR EQUATIONS SOLVERS

ROUTINE | DESCRIPTION |
---|---|

LSLZD | Solves a complex sparse Hermitian positive definite system of linear equations by Gaussian elimination. |

LNFZD | Computes the numerical Cholesky factorization of a sparse Hermitian matrix A. |

LFSZD | Solves a complex sparse Hermitian positive definite system of linear equations, given the Cholesky factorization of the coefficient matrix. |

### REAL TOEPLITZ MATRICES IN TOEPLITZ STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSLTO | Solves a real Toeplitz linear system. |

### COMPLEX TOEPLITZ MATRICES IN TOEPLITZ STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSLTC | Solves a complex Toeplitz linear system. |

### COMPLEX CIRCULAR MATRICES IN CIRCULANT STORAGE MODE

ROUTINE | DESCRIPTION |
---|---|

LSLCC | Solves a complex circulant linear system. |

### ITERATIVE METHODS

ROUTINE | DESCRIPTION |
---|---|

PCGRC | Solves a real symmetric definite linear system using a preconditioned conjugate gradient method with reverse communication. |

JCGRC | Solves a real symmetric definite linear system using the Jacobi-preconditioned conjugate gradient method with reverse communication. |

GMRES | Uses GMRES with reverse communication to generate an approximate solution of Ax = b. |

## LINEAR LEAST SQUARES AND MATRIX FACTORIZATION

### LEAST SQUARES, QR DECOMPOSITION AND GENERALIZED INVERSE LEAST SQUARES

ROUTINE | DESCRIPTION |
---|---|

LSQRR | Solves a linear least-squares problem without iterative refinement. |

LQRRV | Computes the least-squares solution using Householder transformations applied in blocked form. |

LSBRR | Solves a linear least-squares problem with iterative refinement. |

LCLSQ | Solves a linear least-squares problem with linear constraints. |

LQRRR | Computes the QR decomposition, AP = QR, using Householder transformations. |

LQERR | Accumulate the orthogonal matrix Q from its factored form given the QR factorization of a rectangular matrix A. |

LQRSL | Computes the coordinate transformation, projection, and complete the solution of the least-squares problem Ax = b. |

LUPQR | Computes an updated QR factorization after the rank-one matrix αxy is added.^{T} |

### CHOLESKY FACTORIZATION

ROUTINE | DESCRIPTION |
---|---|

LCHRG | Computes the Cholesky decomposition of a symmetric positive semidefinite matrix with optional column pivoting. |

LUPCH | Updates the R Cholesky factorization of a real symmetric positive definite matrix after a rank-one matrix is added.^{T}R |

LDNCH | Downdates the RCholesky factorization of a real symmetric positive definite matrix after a rank-one matrix is removed.^{T}R |

### SINGULAR VALUE DECOMPOSITIONS

ROUTINE | DESCRIPTION |
---|---|

LSVRR | Computes the singular value decomposition of a real matrix. |

LSVCR | Computes the singular value decomposition of a complex matrix. |

LSGRR | Computes the generalized inverse of a real matrix. |