Category D. Linear Algebra

D1. Elementary vector and matrix operations D1. Elementary vector and matrix operations D2. Solution of systems of linear equations D3. Determinants D4. Eigenvalues, eigenvectors D5. QR decomposition, Gram-Schmidt orthogonalization D6. Singular value decomposition D7. Update matrix decompositions D9. Overdetermined or underdetermined systems of equations, singular systems, pseudo-inverses D1A. Elementary vector operations D1A2. Minimum and maximum components ISAMAX-S Find the smallest index of that component of a vector IDAMAX-D having the maximum magnitude. ICAMAX-C D1A3. Norm D1A3A. L-1 (sum of magnitudes) SASUM-S Compute the sum of the magnitudes of the elements of a DASUM-D vector. SCASUM-C D1A3B. L-2 (Euclidean norm) SNRM2-S Compute the Euclidean length (L2 norm) of a vector. DNRM2-D SCNRM2-C D1A4. Dot product (inner product) CDOTC-C Dot product of two complex vectors using the complex conjugate of the first vector. DQDOTA-D Compute the inner product of two vectors with extended precision accumulation and result. DQDOTI-D Compute the inner product of two vectors with extended precision accumulation and result. DSDOT-D Compute the inner product of two vectors with extended DCDOT-C precision accumulation and result. SDOT-S Compute the inner product of two vectors. DDOT-D CDOTU-C SDSDOT-S Compute the inner product of two vectors with extended CDCDOT-C precision accumulation. D1A5. Copy or exchange (swap) ICOPY-S Copy a vector. DCOPY-D CCOPY-C ICOPY-I SCOPY-S Copy a vector. DCOPY-D CCOPY-C ICOPY-I SCOPYM-S Copy the negative of a vector to a vector. DCOPYM-D SSWAP-S Interchange two vectors. DSWAP-D CSWAP-C ISWAP-I D1A6. Multiplication by scalar CSSCAL-C Scale a complex vector. SSCAL-S Multiply a vector by a constant. DSCAL-D CSCAL-C D1A7. Triad (a*x+y for vectors x,y and scalar a) SAXPY-S Compute a constant times a vector plus a vector. DAXPY-D CAXPY-C D1A8. Elementary rotation (Givens transformation) SROT-S Apply a plane Givens rotation. DROT-D CSROT-C SROTM-S Apply a modified Givens transformation. DROTM-D D1B. Elementary matrix operations D1B4. Multiplication by vector CHPR-C Perform the hermitian rank 1 operation. DGER-D Perform the rank 1 operation. DSPR-D Perform the symmetric rank 1 operation. DSYR-D Perform the symmetric rank 1 operation. SGBMV-S Multiply a real vector by a real general band matrix. DGBMV-D CGBMV-C SGEMV-S Multiply a real vector by a real general matrix. DGEMV-D CGEMV-C SGER-S Perform rank 1 update of a real general matrix. CGERC-C Perform conjugated rank 1 update of a complex general SGERC-S matrix. DGERC-D CGERU-C Perform unconjugated rank 1 update of a complex general SGERU-S matrix. DGERU-D CHBMV-C Multiply a complex vector by a complex Hermitian band SHBMV-S matrix. DHBMV-D CHEMV-C Multiply a complex vector by a complex Hermitian matrix. SHEMV-S DHEMV-D CHER-C Perform Hermitian rank 1 update of a complex Hermitian SHER-S matrix. DHER-D CHER2-C Perform Hermitian rank 2 update of a complex Hermitian SHER2-S matrix. DHER2-D CHPMV-C Perform the matrix-vector operation. SHPMV-S DHPMV-D CHPR2-C Perform the hermitian rank 2 operation. SHPR2-S DHPR2-D SSBMV-S Multiply a real vector by a real symmetric band matrix. DSBMV-D CSBMV-C SSDI-S Diagonal Matrix Vector Multiply. DSDI-D Routine to calculate the product X = DIAG*B, where DIAG is a diagonal matrix. SSMTV-S SLAP Column Format Sparse Matrix Transpose Vector Product. DSMTV-D Routine to calculate the sparse matrix vector product: Y = A'*X, where ' denotes transpose. SSMV-S SLAP Column Format Sparse Matrix Vector Product. DSMV-D Routine to calculate the sparse matrix vector product: Y = A*X. SSPMV-S Perform the matrix-vector operation. DSPMV-D CSPMV-C SSPR-S Performs the symmetric rank 1 operation. SSPR2-S Perform the symmetric rank 2 operation. DSPR2-D CSPR2-C SSYMV-S Multiply a real vector by a real symmetric matrix. DSYMV-D CSYMV-C SSYR-S Perform symmetric rank 1 update of a real symmetric matrix. SSYR2-S Perform symmetric rank 2 update of a real symmetric matrix. DSYR2-D CSYR2-C STBMV-S Multiply a real vector by a real triangular band matrix. DTBMV-D CTBMV-C STBSV-S Solve a real triangular banded system of linear equations. DTBSV-D CTBSV-C STPMV-S Perform one of the matrix-vector operations. DTPMV-D CTPMV-C STPSV-S Solve one of the systems of equations. DTPSV-D CTPSV-C STRMV-S Multiply a real vector by a real triangular matrix. DTRMV-D CTRMV-C STRSV-S Solve a real triangular system of linear equations. DTRSV-D CTRSV-C D1B6. Multiplication SGEMM-S Multiply a real general matrix by a real general matrix. DGEMM-D CGEMM-C CHEMM-C Multiply a complex general matrix by a complex Hermitian SHEMM-S matrix. DHEMM-D CHER2K-C Perform Hermitian rank 2k update of a complex. SHER2-S DHER2-D CHER2-C CHERK-C Perform Hermitian rank k update of a complex Hermitian SHERK-S matrix. DHERK-D SSYMM-S Multiply a real general matrix by a real symmetric matrix. DSYMM-D CSYMM-C DSYR2K-D Perform one of the symmetric rank 2k operations. SSYR2-S DSYR2-D CSYR2-C SSYRK-S Perform symmetric rank k update of a real symmetric matrix. DSYRK-D CSYRK-C STRMM-S Multiply a real general matrix by a real triangular matrix. DTRMM-D CTRMM-C STRSM-S Solve a real triangular system of equations with multiple DTRSM-D right-hand sides. CTRSM-C D1B9. Storage mode conversion SS2Y-S SLAP Triad to SLAP Column Format Converter. DS2Y-D Routine to convert from the SLAP Triad to SLAP Column format. D1B10. Elementary rotation (Givens transformation) CSROT-C Apply a plane Givens rotation. SROT-S DROT-D SROTG-S Construct a plane Givens rotation. DROTG-D CROTG-C SROTMG-S Construct a modified Givens transformation. DROTMG-D D2. Solution of systems of linear equations (including inversion, LU and related decompositions) D2A. Real nonsymmetric matrices D2A1. General SGECO-S Factor a matrix using Gaussian elimination and estimate DGECO-D the condition number of the matrix. CGECO-C SGEDI-S Compute the determinant and inverse of a matrix using the DGEDI-D factors computed by SGECO or SGEFA. CGEDI-C SGEFA-S Factor a matrix using Gaussian elimination. DGEFA-D CGEFA-C SGEFS-S Solve a general system of linear equations. DGEFS-D CGEFS-C SGEIR-S Solve a general system of linear equations. Iterative CGEIR-C refinement is used to obtain an error estimate. SGESL-S Solve the real system A*X=B or TRANS(A)*X=B using the DGESL-D factors of SGECO or SGEFA. CGESL-C SQRSL-S Apply the output of SQRDC to compute coordinate transfor- DQRSL-D mations, projections, and least squares solutions. CQRSL-C D2A2. Banded SGBCO-S Factor a band matrix by Gaussian elimination and DGBCO-D estimate the condition number of the matrix. CGBCO-C SGBFA-S Factor a band matrix using Gaussian elimination. DGBFA-D CGBFA-C SGBSL-S Solve the real band system A*X=B or TRANS(A)*X=B using DGBSL-D the factors computed by SGBCO or SGBFA. CGBSL-C SNBCO-S Factor a band matrix using Gaussian elimination and DNBCO-D estimate the condition number. CNBCO-C SNBFA-S Factor a real band matrix by elimination. DNBFA-D CNBFA-C SNBFS-S Solve a general nonsymmetric banded system of linear DNBFS-D equations. CNBFS-C SNBIR-S Solve a general nonsymmetric banded system of linear CNBIR-C equations. Iterative refinement is used to obtain an error estimate. SNBSL-S Solve a real band system using the factors computed by DNBSL-D SNBCO or SNBFA. CNBSL-C D2A2A. Tridiagonal SGTSL-S Solve a tridiagonal linear system. DGTSL-D CGTSL-C D2A3. Triangular SSLI-S SLAP MSOLVE for Lower Triangle Matrix. DSLI-D This routine acts as an interface between the SLAP generic MSOLVE calling convention and the routine that actually -1 computes L B = X. SSLI2-S SLAP Lower Triangle Matrix Backsolve. DSLI2-D Routine to solve a system of the form Lx = b , where L is a lower triangular matrix. STRCO-S Estimate the condition number of a triangular matrix. DTRCO-D CTRCO-C STRDI-S Compute the determinant and inverse of a triangular matrix. DTRDI-D CTRDI-C STRSL-S Solve a system of the form T*X=B or TRANS(T)*X=B, where DTRSL-D T is a triangular matrix. CTRSL-C D2A4. Sparse SBCG-S Preconditioned BiConjugate Gradient Sparse Ax = b Solver. DBCG-D Routine to solve a Non-Symmetric linear system Ax = b using the Preconditioned BiConjugate Gradient method. SCGN-S Preconditioned CG Sparse Ax=b Solver for Normal Equations. DCGN-D Routine to solve a general linear system Ax = b using the Preconditioned Conjugate Gradient method applied to the normal equations AA'y = b, x=A'y. SCGS-S Preconditioned BiConjugate Gradient Squared Ax=b Solver. DCGS-D Routine to solve a Non-Symmetric linear system Ax = b using the Preconditioned BiConjugate Gradient Squared method. SGMRES-S Preconditioned GMRES Iterative Sparse Ax=b Solver. DGMRES-D This routine uses the generalized minimum residual (GMRES) method with preconditioning to solve non-symmetric linear systems of the form: Ax = b. SIR-S Preconditioned Iterative Refinement Sparse Ax = b Solver. DIR-D Routine to solve a general linear system Ax = b using iterative refinement with a matrix splitting. SLPDOC-S Sparse Linear Algebra Package Version 2.0.2 Documentation. DLPDOC-D Routines to solve large sparse symmetric and nonsymmetric positive definite linear systems, Ax = b, using precondi- tioned iterative methods. SOMN-S Preconditioned Orthomin Sparse Iterative Ax=b Solver. DOMN-D Routine to solve a general linear system Ax = b using the Preconditioned Orthomin method. SSDBCG-S Diagonally Scaled BiConjugate Gradient Sparse Ax=b Solver. DSDBCG-D Routine to solve a linear system Ax = b using the BiConjugate Gradient method with diagonal scaling. SSDCGN-S Diagonally Scaled CG Sparse Ax=b Solver for Normal Eqn's. DSDCGN-D Routine to solve a general linear system Ax = b using diagonal scaling with the Conjugate Gradient method applied to the the normal equations, viz., AA'y = b, where x = A'y. SSDCGS-S Diagonally Scaled CGS Sparse Ax=b Solver. DSDCGS-D Routine to solve a linear system Ax = b using the BiConjugate Gradient Squared method with diagonal scaling. SSDGMR-S Diagonally Scaled GMRES Iterative Sparse Ax=b Solver. DSDGMR-D This routine uses the generalized minimum residual (GMRES) method with diagonal scaling to solve possibly non-symmetric linear systems of the form: Ax = b. SSDOMN-S Diagonally Scaled Orthomin Sparse Iterative Ax=b Solver. DSDOMN-D Routine to solve a general linear system Ax = b using the Orthomin method with diagonal scaling. SSGS-S Gauss-Seidel Method Iterative Sparse Ax = b Solver. DSGS-D Routine to solve a general linear system Ax = b using Gauss-Seidel iteration. SSILUR-S Incomplete LU Iterative Refinement Sparse Ax = b Solver. DSILUR-D Routine to solve a general linear system Ax = b using the incomplete LU decomposition with iterative refinement. SSJAC-S Jacobi's Method Iterative Sparse Ax = b Solver. DSJAC-D Routine to solve a general linear system Ax = b using Jacobi iteration. SSLUBC-S Incomplete LU BiConjugate Gradient Sparse Ax=b Solver. DSLUBC-D Routine to solve a linear system Ax = b using the BiConjugate Gradient method with Incomplete LU decomposition preconditioning. SSLUCN-S Incomplete LU CG Sparse Ax=b Solver for Normal Equations. DSLUCN-D Routine to solve a general linear system Ax = b using the incomplete LU decomposition with the Conjugate Gradient method applied to the normal equations, viz., AA'y = b, x = A'y. SSLUCS-S Incomplete LU BiConjugate Gradient Squared Ax=b Solver. DSLUCS-D Routine to solve a linear system Ax = b using the BiConjugate Gradient Squared method with Incomplete LU decomposition preconditioning. SSLUGM-S Incomplete LU GMRES Iterative Sparse Ax=b Solver. DSLUGM-D This routine uses the generalized minimum residual (GMRES) method with incomplete LU factorization for preconditioning to solve possibly non-symmetric linear systems of the form: Ax = b. SSLUOM-S Incomplete LU Orthomin Sparse Iterative Ax=b Solver. DSLUOM-D Routine to solve a general linear system Ax = b using the Orthomin method with Incomplete LU decomposition. D2B. Real symmetric matrices D2B1. General D2B1A. Indefinite SSICO-S Factor a symmetric matrix by elimination with symmetric DSICO-D pivoting and estimate the condition number of the matrix. CHICO-C CSICO-C SSIDI-S Compute the determinant, inertia and inverse of a real DSIDI-D symmetric matrix using the factors from SSIFA. CHIDI-C CSIDI-C SSIFA-S Factor a real symmetric matrix by elimination with DSIFA-D symmetric pivoting. CHIFA-C CSIFA-C SSISL-S Solve a real symmetric system using the factors obtained DSISL-D from SSIFA. CHISL-C CSISL-C SSPCO-S Factor a real symmetric matrix stored in packed form DSPCO-D by elimination with symmetric pivoting and estimate the CHPCO-C condition number of the matrix. CSPCO-C SSPDI-S Compute the determinant, inertia, inverse of a real DSPDI-D symmetric matrix stored in packed form using the factors CHPDI-C from SSPFA. CSPDI-C SSPFA-S Factor a real symmetric matrix stored in packed form by DSPFA-D elimination with symmetric pivoting. CHPFA-C CSPFA-C SSPSL-S Solve a real symmetric system using the factors obtained DSPSL-D from SSPFA. CHPSL-C CSPSL-C D2B1B. Positive definite SCHDC-S Compute the Cholesky decomposition of a positive definite DCHDC-D matrix. A pivoting option allows the user to estimate the CCHDC-C condition number of a positive definite matrix or determine the rank of a positive semidefinite matrix. SPOCO-S Factor a real symmetric positive definite matrix DPOCO-D and estimate the condition number of the matrix. CPOCO-C SPODI-S Compute the determinant and inverse of a certain real DPODI-D symmetric positive definite matrix using the factors CPODI-C computed by SPOCO, SPOFA or SQRDC. SPOFA-S Factor a real symmetric positive definite matrix. DPOFA-D CPOFA-C SPOFS-S Solve a positive definite symmetric system of linear DPOFS-D equations. CPOFS-C SPOIR-S Solve a positive definite symmetric system of linear CPOIR-C equations. Iterative refinement is used to obtain an error estimate. SPOSL-S Solve the real symmetric positive definite linear system DPOSL-D using the factors computed by SPOCO or SPOFA. CPOSL-C SPPCO-S Factor a symmetric positive definite matrix stored in DPPCO-D packed form and estimate the condition number of the CPPCO-C matrix. SPPDI-S Compute the determinant and inverse of a real symmetric DPPDI-D positive definite matrix using factors from SPPCO or SPPFA. CPPDI-C SPPFA-S Factor a real symmetric positive definite matrix stored in DPPFA-D packed form. CPPFA-C SPPSL-S Solve the real symmetric positive definite system using DPPSL-D the factors computed by SPPCO or SPPFA. CPPSL-C D2B2. Positive definite banded SPBCO-S Factor a real symmetric positive definite matrix stored in DPBCO-D band form and estimate the condition number of the matrix. CPBCO-C SPBFA-S Factor a real symmetric positive definite matrix stored in DPBFA-D band form. CPBFA-C SPBSL-S Solve a real symmetric positive definite band system DPBSL-D using the factors computed by SPBCO or SPBFA. CPBSL-C D2B2A. Tridiagonal SPTSL-S Solve a positive definite tridiagonal linear system. DPTSL-D CPTSL-C D2B4. Sparse SBCG-S Preconditioned BiConjugate Gradient Sparse Ax = b Solver. DBCG-D Routine to solve a Non-Symmetric linear system Ax = b using the Preconditioned BiConjugate Gradient method. SCG-S Preconditioned Conjugate Gradient Sparse Ax=b Solver. DCG-D Routine to solve a symmetric positive definite linear system Ax = b using the Preconditioned Conjugate Gradient method. SCGN-S Preconditioned CG Sparse Ax=b Solver for Normal Equations. DCGN-D Routine to solve a general linear system Ax = b using the Preconditioned Conjugate Gradient method applied to the normal equations AA'y = b, x=A'y. SCGS-S Preconditioned BiConjugate Gradient Squared Ax=b Solver. DCGS-D Routine to solve a Non-Symmetric linear system Ax = b using the Preconditioned BiConjugate Gradient Squared method. SGMRES-S Preconditioned GMRES Iterative Sparse Ax=b Solver. DGMRES-D This routine uses the generalized minimum residual (GMRES) method with preconditioning to solve non-symmetric linear systems of the form: Ax = b. SIR-S Preconditioned Iterative Refinement Sparse Ax = b Solver. DIR-D Routine to solve a general linear system Ax = b using iterative refinement with a matrix splitting. SLPDOC-S Sparse Linear Algebra Package Version 2.0.2 Documentation. DLPDOC-D Routines to solve large sparse symmetric and nonsymmetric positive definite linear systems, Ax = b, using precondi- tioned iterative methods. SOMN-S Preconditioned Orthomin Sparse Iterative Ax=b Solver. DOMN-D Routine to solve a general linear system Ax = b using the Preconditioned Orthomin method. SSDBCG-S Diagonally Scaled BiConjugate Gradient Sparse Ax=b Solver. DSDBCG-D Routine to solve a linear system Ax = b using the BiConjugate Gradient method with diagonal scaling. SSDCG-S Diagonally Scaled Conjugate Gradient Sparse Ax=b Solver. DSDCG-D Routine to solve a symmetric positive definite linear system Ax = b using the Preconditioned Conjugate Gradient method. The preconditioner is diagonal scaling. SSDCGN-S Diagonally Scaled CG Sparse Ax=b Solver for Normal Eqn's. DSDCGN-D Routine to solve a general linear system Ax = b using diagonal scaling with the Conjugate Gradient method applied to the the normal equations, viz., AA'y = b, where x = A'y. SSDCGS-S Diagonally Scaled CGS Sparse Ax=b Solver. DSDCGS-D Routine to solve a linear system Ax = b using the BiConjugate Gradient Squared method with diagonal scaling. SSDGMR-S Diagonally Scaled GMRES Iterative Sparse Ax=b Solver. DSDGMR-D This routine uses the generalized minimum residual (GMRES) method with diagonal scaling to solve possibly non-symmetric linear systems of the form: Ax = b. SSDOMN-S Diagonally Scaled Orthomin Sparse Iterative Ax=b Solver. DSDOMN-D Routine to solve a general linear system Ax = b using the Orthomin method with diagonal scaling. SSGS-S Gauss-Seidel Method Iterative Sparse Ax = b Solver. DSGS-D Routine to solve a general linear system Ax = b using Gauss-Seidel iteration. SSICCG-S Incomplete Cholesky Conjugate Gradient Sparse Ax=b Solver. DSICCG-D Routine to solve a symmetric positive definite linear system Ax = b using the incomplete Cholesky Preconditioned Conjugate Gradient method. SSILUR-S Incomplete LU Iterative Refinement Sparse Ax = b Solver. DSILUR-D Routine to solve a general linear system Ax = b using the incomplete LU decomposition with iterative refinement. SSJAC-S Jacobi's Method Iterative Sparse Ax = b Solver. DSJAC-D Routine to solve a general linear system Ax = b using Jacobi iteration. SSLUBC-S Incomplete LU BiConjugate Gradient Sparse Ax=b Solver. DSLUBC-D Routine to solve a linear system Ax = b using the BiConjugate Gradient method with Incomplete LU decomposition preconditioning. SSLUCN-S Incomplete LU CG Sparse Ax=b Solver for Normal Equations. DSLUCN-D Routine to solve a general linear system Ax = b using the incomplete LU decomposition with the Conjugate Gradient method applied to the normal equations, viz., AA'y = b, x = A'y. SSLUCS-S Incomplete LU BiConjugate Gradient Squared Ax=b Solver. DSLUCS-D Routine to solve a linear system Ax = b using the BiConjugate Gradient Squared method with Incomplete LU decomposition preconditioning. SSLUGM-S Incomplete LU GMRES Iterative Sparse Ax=b Solver. DSLUGM-D This routine uses the generalized minimum residual (GMRES) method with incomplete LU factorization for preconditioning to solve possibly non-symmetric linear systems of the form: Ax = b. SSLUOM-S Incomplete LU Orthomin Sparse Iterative Ax=b Solver. DSLUOM-D Routine to solve a general linear system Ax = b using the Orthomin method with Incomplete LU decomposition. D2C. Complex non-Hermitian matrices D2C1. General CGECO-C Factor a matrix using Gaussian elimination and estimate SGECO-S the condition number of the matrix. DGECO-D CGEDI-C Compute the determinant and inverse of a matrix using the SGEDI-S factors computed by CGECO or CGEFA. DGEDI-D CGEFA-C Factor a matrix using Gaussian elimination. SGEFA-S DGEFA-D CGEFS-C Solve a general system of linear equations. SGEFS-S DGEFS-D CGEIR-C Solve a general system of linear equations. Iterative SGEIR-S refinement is used to obtain an error estimate. CGESL-C Solve the complex system A*X=B or CTRANS(A)*X=B using the SGESL-S factors computed by CGECO or CGEFA. DGESL-D CQRSL-C Apply the output of CQRDC to compute coordinate transfor- SQRSL-S mations, projections, and least squares solutions. DQRSL-D CSICO-C Factor a complex symmetric matrix by elimination with SSICO-S symmetric pivoting and estimate the condition number of the DSICO-D matrix. CHICO-C CSIDI-C Compute the determinant and inverse of a complex symmetric SSIDI-S matrix using the factors from CSIFA. DSIDI-D CHIDI-C CSIFA-C Factor a complex symmetric matrix by elimination with SSIFA-S symmetric pivoting. DSIFA-D CHIFA-C CSISL-C Solve a complex symmetric system using the factors obtained SSISL-S from CSIFA. DSISL-D CHISL-C CSPCO-C Factor a complex symmetric matrix stored in packed form SSPCO-S by elimination with symmetric pivoting and estimate the DSPCO-D condition number of the matrix. CHPCO-C CSPDI-C Compute the determinant and inverse of a complex symmetric SSPDI-S matrix stored in packed form using the factors from CSPFA. DSPDI-D CHPDI-C CSPFA-C Factor a complex symmetric matrix stored in packed form by SSPFA-S elimination with symmetric pivoting. DSPFA-D CHPFA-C CSPSL-C Solve a complex symmetric system using the factors obtained SSPSL-S from CSPFA. DSPSL-D CHPSL-C D2C2. Banded CGBCO-C Factor a band matrix by Gaussian elimination and SGBCO-S estimate the condition number of the matrix. DGBCO-D CGBFA-C Factor a band matrix using Gaussian elimination. SGBFA-S DGBFA-D CGBSL-C Solve the complex band system A*X=B or CTRANS(A)*X=B using SGBSL-S the factors computed by CGBCO or CGBFA. DGBSL-D CNBCO-C Factor a band matrix using Gaussian elimination and SNBCO-S estimate the condition number. DNBCO-D CNBFA-C Factor a band matrix by elimination. SNBFA-S DNBFA-D CNBFS-C Solve a general nonsymmetric banded system of linear SNBFS-S equations. DNBFS-D CNBIR-C Solve a general nonsymmetric banded system of linear SNBIR-S equations. Iterative refinement is used to obtain an error estimate. CNBSL-C Solve a complex band system using the factors computed by SNBSL-S CNBCO or CNBFA. DNBSL-D D2C2A. Tridiagonal CGTSL-C Solve a tridiagonal linear system. SGTSL-S DGTSL-D D2C3. Triangular CTRCO-C Estimate the condition number of a triangular matrix. STRCO-S DTRCO-D CTRDI-C Compute the determinant and inverse of a triangular matrix. STRDI-S DTRDI-D CTRSL-C Solve a system of the form T*X=B or CTRANS(T)*X=B, where STRSL-S T is a triangular matrix. Here CTRANS(T) is the conjugate DTRSL-D transpose. D2D. Complex Hermitian matrices D2D1. General D2D1A. Indefinite CHICO-C Factor a complex Hermitian matrix by elimination with sym- SSICO-S metric pivoting and estimate the condition of the matrix. DSICO-D CSICO-C CHIDI-C Compute the determinant, inertia and inverse of a complex SSIDI-S Hermitian matrix using the factors obtained from CHIFA. DSISI-D CSIDI-C CHIFA-C Factor a complex Hermitian matrix by elimination SSIFA-S (symmetric pivoting). DSIFA-D CSIFA-C CHISL-C Solve the complex Hermitian system using factors obtained SSISL-S from CHIFA. DSISL-D CSISL-C CHPCO-C Factor a complex Hermitian matrix stored in packed form by SSPCO-S elimination with symmetric pivoting and estimate the DSPCO-D condition number of the matrix. CSPCO-C CHPDI-C Compute the determinant, inertia and inverse of a complex SSPDI-S Hermitian matrix stored in packed form using the factors DSPDI-D obtained from CHPFA. DSPDI-C CHPFA-C Factor a complex Hermitian matrix stored in packed form by SSPFA-S elimination with symmetric pivoting. DSPFA-D DSPFA-C CHPSL-C Solve a complex Hermitian system using factors obtained SSPSL-S from CHPFA. DSPSL-D CSPSL-C D2D1B. Positive definite CCHDC-C Compute the Cholesky decomposition of a positive definite SCHDC-S matrix. A pivoting option allows the user to estimate the DCHDC-D condition number of a positive definite matrix or determine the rank of a positive semidefinite matrix. CPOCO-C Factor a complex Hermitian positive definite matrix SPOCO-S and estimate the condition number of the matrix. DPOCO-D CPODI-C Compute the determinant and inverse of a certain complex SPODI-S Hermitian positive definite matrix using the factors DPODI-D computed by CPOCO, CPOFA, or CQRDC. CPOFA-C Factor a complex Hermitian positive definite matrix. SPOFA-S DPOFA-D CPOFS-C Solve a positive definite symmetric complex system of SPOFS-S linear equations. DPOFS-D CPOIR-C Solve a positive definite Hermitian system of linear SPOIR-S equations. Iterative refinement is used to obtain an error estimate. CPOSL-C Solve the complex Hermitian positive definite linear system SPOSL-S using the factors computed by CPOCO or CPOFA. DPOSL-D CPPCO-C Factor a complex Hermitian positive definite matrix stored SPPCO-S in packed form and estimate the condition number of the DPPCO-D matrix. CPPDI-C Compute the determinant and inverse of a complex Hermitian SPPDI-S positive definite matrix using factors from CPPCO or CPPFA. DPPDI-D CPPFA-C Factor a complex Hermitian positive definite matrix stored SPPFA-S in packed form. DPPFA-D CPPSL-C Solve the complex Hermitian positive definite system using SPPSL-S the factors computed by CPPCO or CPPFA. DPPSL-D D2D2. Positive definite banded CPBCO-C Factor a complex Hermitian positive definite matrix stored SPBCO-S in band form and estimate the condition number of the DPBCO-D matrix. CPBFA-C Factor a complex Hermitian positive definite matrix stored SPBFA-S in band form. DPBFA-D CPBSL-C Solve the complex Hermitian positive definite band system SPBSL-S using the factors computed by CPBCO or CPBFA. DPBSL-D D2D2A. Tridiagonal CPTSL-C Solve a positive definite tridiagonal linear system. SPTSL-S DPTSL-D D2E. Associated operations (e.g., matrix reorderings) SLLTI2-S SLAP Backsolve routine for LDL' Factorization. DLLTI2-D Routine to solve a system of the form L*D*L' X = B, where L is a unit lower triangular matrix and D is a diagonal matrix and ' means transpose. SS2LT-S Lower Triangle Preconditioner SLAP Set Up. DS2LT-D Routine to store the lower triangle of a matrix stored in the SLAP Column format. SSD2S-S Diagonal Scaling Preconditioner SLAP Normal Eqns Set Up. DSD2S-D Routine to compute the inverse of the diagonal of the matrix A*A', where A is stored in SLAP-Column format. SSDS-S Diagonal Scaling Preconditioner SLAP Set Up. DSDS-D Routine to compute the inverse of the diagonal of a matrix stored in the SLAP Column format. SSDSCL-S Diagonal Scaling of system Ax = b. DSDSCL-D This routine scales (and unscales) the system Ax = b by symmetric diagonal scaling. SSICS-S Incompl. Cholesky Decomposition Preconditioner SLAP Set Up. DSICS-D Routine to generate the Incomplete Cholesky decomposition, L*D*L-trans, of a symmetric positive definite matrix, A, which is stored in SLAP Column format. The unit lower triangular matrix L is stored by rows, and the inverse of the diagonal matrix D is stored. SSILUS-S Incomplete LU Decomposition Preconditioner SLAP Set Up. DSILUS-D Routine to generate the incomplete LDU decomposition of a matrix. The unit lower triangular factor L is stored by rows and the unit upper triangular factor U is stored by columns. The inverse of the diagonal matrix D is stored. No fill in is allowed. SSLLTI-S SLAP MSOLVE for LDL' (IC) Factorization. DSLLTI-D This routine acts as an interface between the SLAP generic MSOLVE calling convention and the routine that actually -1 computes (LDL') B = X. SSLUI-S SLAP MSOLVE for LDU Factorization. DSLUI-D This routine acts as an interface between the SLAP generic MSOLVE calling convention and the routine that actually -1 computes (LDU) B = X. SSLUI2-S SLAP Backsolve for LDU Factorization. DSLUI2-D Routine to solve a system of the form L*D*U X = B, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix. SSLUI4-S SLAP Backsolve for LDU Factorization. DSLUI4-D Routine to solve a system of the form (L*D*U)' X = B, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix and ' denotes transpose. SSLUTI-S SLAP MTSOLV for LDU Factorization. DSLUTI-D This routine acts as an interface between the SLAP generic MTSOLV calling convention and the routine that actually -T computes (LDU) B = X. SSMMI2-S SLAP Backsolve for LDU Factorization of Normal Equations. DSMMI2-D To solve a system of the form (L*D*U)*(L*D*U)' X = B, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix and ' denotes transpose. SSMMTI-S SLAP MSOLVE for LDU Factorization of Normal Equations. DSMMTI-D This routine acts as an interface between the SLAP generic MMTSLV calling convention and the routine that actually -1 computes [(LDU)*(LDU)'] B = X. D3. Determinants D3A. Real nonsymmetric matrices D3A1. General SGEDI-S Compute the determinant and inverse of a matrix using the DGEDI-D factors computed by SGECO or SGEFA. CGEDI-C D3A2. Banded SGBDI-S Compute the determinant of a band matrix using the factors DGBDI-D computed by SGBCO or SGBFA. CGBDI-C SNBDI-S Compute the determinant of a band matrix using the factors DNBDI-D computed by SNBCO or SNBFA. CNBDI-C D3A3. Triangular STRDI-S Compute the determinant and inverse of a triangular matrix. DTRDI-D CTRDI-C D3B. Real symmetric matrices D3B1. General D3B1A. Indefinite SSIDI-S Compute the determinant, inertia and inverse of a real DSIDI-D symmetric matrix using the factors from SSIFA. CHIDI-C CSIDI-C SSPDI-S Compute the determinant, inertia, inverse of a real DSPDI-D symmetric matrix stored in packed form using the factors CHPDI-C from SSPFA. CSPDI-C D3B1B. Positive definite SPODI-S Compute the determinant and inverse of a certain real DPODI-D symmetric positive definite matrix using the factors CPODI-C computed by SPOCO, SPOFA or SQRDC. SPPDI-S Compute the determinant and inverse of a real symmetric DPPDI-D positive definite matrix using factors from SPPCO or SPPFA. CPPDI-C D3B2. Positive definite banded SPBDI-S Compute the determinant of a symmetric positive definite DPBDI-D band matrix using the factors computed by SPBCO or SPBFA. CPBDI-C D3C. Complex non-Hermitian matrices D3C1. General CGEDI-C Compute the determinant and inverse of a matrix using the SGEDI-S factors computed by CGECO or CGEFA. DGEDI-D CSIDI-C Compute the determinant and inverse of a complex symmetric SSIDI-S matrix using the factors from CSIFA. DSIDI-D CHIDI-C CSPDI-C Compute the determinant and inverse of a complex symmetric SSPDI-S matrix stored in packed form using the factors from CSPFA. DSPDI-D CHPDI-C D3C2. Banded CGBDI-C Compute the determinant of a complex band matrix using the SGBDI-S factors from CGBCO or CGBFA. DGBDI-D CNBDI-C Compute the determinant of a band matrix using the factors SNBDI-S computed by CNBCO or CNBFA. DNBDI-D D3C3. Triangular CTRDI-C Compute the determinant and inverse of a triangular matrix. STRDI-S DTRDI-D D3D. Complex Hermitian matrices D3D1. General D3D1A. Indefinite CHIDI-C Compute the determinant, inertia and inverse of a complex SSIDI-S Hermitian matrix using the factors obtained from CHIFA. DSISI-D CSIDI-C CHPDI-C Compute the determinant, inertia and inverse of a complex SSPDI-S Hermitian matrix stored in packed form using the factors DSPDI-D obtained from CHPFA. DSPDI-C D3D1B. Positive definite CPODI-C Compute the determinant and inverse of a certain complex SPODI-S Hermitian positive definite matrix using the factors DPODI-D computed by CPOCO, CPOFA, or CQRDC. CPPDI-C Compute the determinant and inverse of a complex Hermitian SPPDI-S positive definite matrix using factors from CPPCO or CPPFA. DPPDI-D D3D2. Positive definite banded CPBDI-C Compute the determinant of a complex Hermitian positive SPBDI-S definite band matrix using the factors computed by CPBCO or DPBDI-D CPBFA. D4. Eigenvalues, eigenvectors EISDOC-A Documentation for EISPACK, a collection of subprograms for solving matrix eigen-problems. D4A. Ordinary eigenvalue problems (Ax = (lambda) * x) D4A1. Real symmetric RS-S Compute the eigenvalues and, optionally, the eigenvectors CH-C of a real symmetric matrix. RSP-S Compute the eigenvalues and, optionally, the eigenvectors of a real symmetric matrix packed into a one dimensional array. SSIEV-S Compute the eigenvalues and, optionally, the eigenvectors CHIEV-C of a real symmetric matrix. SSPEV-S Compute the eigenvalues and, optionally, the eigenvectors of a real symmetric matrix stored in packed form. D4A2. Real nonsymmetric RG-S Compute the eigenvalues and, optionally, the eigenvectors CG-C of a real general matrix. SGEEV-S Compute the eigenvalues and, optionally, the eigenvectors CGEEV-C of a real general matrix. D4A3. Complex Hermitian CH-C Compute the eigenvalues and, optionally, the eigenvectors RS-S of a complex Hermitian matrix. CHIEV-C Compute the eigenvalues and, optionally, the eigenvectors SSIEV-S of a complex Hermitian matrix. D4A4. Complex non-Hermitian CG-C Compute the eigenvalues and, optionally, the eigenvectors RG-S of a complex general matrix. CGEEV-C Compute the eigenvalues and, optionally, the eigenvectors SGEEV-S of a complex general matrix. D4A5. Tridiagonal BISECT-S Compute the eigenvalues of a symmetric tridiagonal matrix in a given interval using Sturm sequencing. IMTQL1-S Compute the eigenvalues of a symmetric tridiagonal matrix using the implicit QL method. IMTQL2-S Compute the eigenvalues and eigenvectors of a symmetric tridiagonal matrix using the implicit QL method. IMTQLV-S Compute the eigenvalues of a symmetric tridiagonal matrix using the implicit QL method. Eigenvectors may be computed later. RATQR-S Compute the largest or smallest eigenvalues of a symmetric tridiagonal matrix using the rational QR method with Newton correction. RST-S Compute the eigenvalues and, optionally, the eigenvectors of a real symmetric tridiagonal matrix. RT-S Compute the eigenvalues and eigenvectors of a special real tridiagonal matrix. TQL1-S Compute the eigenvalues of symmetric tridiagonal matrix by the QL method. TQL2-S Compute the eigenvalues and eigenvectors of symmetric tridiagonal matrix. TQLRAT-S Compute the eigenvalues of symmetric tridiagonal matrix using a rational variant of the QL method. TRIDIB-S Compute the eigenvalues of a symmetric tridiagonal matrix in a given interval using Sturm sequencing. TSTURM-S Find those eigenvalues of a symmetric tridiagonal matrix in a given interval and their associated eigenvectors by Sturm sequencing. D4A6. Banded BQR-S Compute some of the eigenvalues of a real symmetric matrix using the QR method with shifts of origin. RSB-S Compute the eigenvalues and, optionally, the eigenvectors of a symmetric band matrix. D4B. Generalized eigenvalue problems (e.g., Ax = (lambda)*Bx) D4B1. Real symmetric RSG-S Compute the eigenvalues and, optionally, the eigenvectors of a symmetric generalized eigenproblem. RSGAB-S Compute the eigenvalues and, optionally, the eigenvectors of a symmetric generalized eigenproblem. RSGBA-S Compute the eigenvalues and, optionally, the eigenvectors of a symmetric generalized eigenproblem. D4B2. Real general RGG-S Compute the eigenvalues and eigenvectors for a real generalized eigenproblem. D4C. Associated operations D4C1. Transform problem D4C1A. Balance matrix BALANC-S Balance a real general matrix and isolate eigenvalues CBAL-C whenever possible. D4C1B. Reduce to compact form D4C1B1. Tridiagonal BANDR-S Reduce a real symmetric band matrix to symmetric tridiagonal matrix and, optionally, accumulate orthogonal similarity transformations. HTRID3-S Reduce a complex Hermitian (packed) matrix to a real symmetric tridiagonal matrix by unitary similarity transformations. HTRIDI-S Reduce a complex Hermitian matrix to a real symmetric tridiagonal matrix using unitary similarity transformations. TRED1-S Reduce a real symmetric matrix to symmetric tridiagonal matrix using orthogonal similarity transformations. TRED2-S Reduce a real symmetric matrix to a symmetric tridiagonal matrix using and accumulating orthogonal transformations. TRED3-S Reduce a real symmetric matrix stored in packed form to symmetric tridiagonal matrix using orthogonal transformations. D4C1B2. Hessenberg ELMHES-S Reduce a real general matrix to upper Hessenberg form COMHES-C using stabilized elementary similarity transformations. ORTHES-S Reduce a real general matrix to upper Hessenberg form CORTH-C using orthogonal similarity transformations. D4C1B3. Other QZHES-S The first step of the QZ algorithm for solving generalized matrix eigenproblems. Accepts a pair of real general matrices and reduces one of them to upper Hessenberg and the other to upper triangular form using orthogonal transformations. Usually followed by QZIT, QZVAL, QZVEC. QZIT-S The second step of the QZ algorithm for generalized eigenproblems. Accepts an upper Hessenberg and an upper triangular matrix and reduces the former to quasi-triangular form while preserving the form of the latter. Usually preceded by QZHES and followed by QZVAL and QZVEC. D4C1C. Standardize problem FIGI-S Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix. FIGI2-S Transforms certain real non-symmetric tridiagonal matrix to symmetric tridiagonal matrix. REDUC-S Reduce a generalized symmetric eigenproblem to a standard symmetric eigenproblem using Cholesky factorization. REDUC2-S Reduce a certain generalized symmetric eigenproblem to a standard symmetric eigenproblem using Cholesky factorization. D4C2. Compute eigenvalues of matrix in compact form D4C2A. Tridiagonal BISECT-S Compute the eigenvalues of a symmetric tridiagonal matrix in a given interval using Sturm sequencing. IMTQL1-S Compute the eigenvalues of a symmetric tridiagonal matrix using the implicit QL method. IMTQL2-S Compute the eigenvalues and eigenvectors of a symmetric tridiagonal matrix using the implicit QL method. IMTQLV-S Compute the eigenvalues of a symmetric tridiagonal matrix using the implicit QL method. Eigenvectors may be computed later. RATQR-S Compute the largest or smallest eigenvalues of a symmetric tridiagonal matrix using the rational QR method with Newton correction. TQL1-S Compute the eigenvalues of symmetric tridiagonal matrix by the QL method. TQL2-S Compute the eigenvalues and eigenvectors of symmetric tridiagonal matrix. TQLRAT-S Compute the eigenvalues of symmetric tridiagonal matrix using a rational variant of the QL method. TRIDIB-S Compute the eigenvalues of a symmetric tridiagonal matrix in a given interval using Sturm sequencing. TSTURM-S Find those eigenvalues of a symmetric tridiagonal matrix in a given interval and their associated eigenvectors by Sturm sequencing. D4C2B. Hessenberg COMLR-C Compute the eigenvalues of a complex upper Hessenberg matrix using the modified LR method. COMLR2-C Compute the eigenvalues and eigenvectors of a complex upper Hessenberg matrix using the modified LR method. HQR-S Compute the eigenvalues of a real upper Hessenberg matrix COMQR-C using the QR method. HQR2-S Compute the eigenvalues and eigenvectors of a real upper COMQR2-C Hessenberg matrix using QR method. INVIT-S Compute the eigenvectors of a real upper Hessenberg CINVIT-C matrix associated with specified eigenvalues by inverse iteration. D4C2C. Other QZVAL-S The third step of the QZ algorithm for generalized eigenproblems. Accepts a pair of real matrices, one in quasi-triangular form and the other in upper triangular form and computes the eigenvalues of the associated eigenproblem. Usually preceded by QZHES, QZIT, and followed by QZVEC. D4C3. Form eigenvectors from eigenvalues BANDV-S Form the eigenvectors of a real symmetric band matrix associated with a set of ordered approximate eigenvalues by inverse iteration. QZVEC-S The optional fourth step of the QZ algorithm for generalized eigenproblems. Accepts a matrix in quasi-triangular form and another in upper triangular and computes the eigenvectors of the triangular problem and transforms them back to the original coordinates Usually preceded by QZHES, QZIT, and QZVAL. TINVIT-S Compute the eigenvectors of symmetric tridiagonal matrix corresponding to specified eigenvalues, using inverse iteration. D4C4. Back transform eigenvectors BAKVEC-S Form the eigenvectors of a certain real non-symmetric tridiagonal matrix from a symmetric tridiagonal matrix output from FIGI. BALBAK-S Form the eigenvectors of a real general matrix from the CBABK2-C eigenvectors of matrix output from BALANC. ELMBAK-S Form the eigenvectors of a real general matrix from the COMBAK-C eigenvectors of the upper Hessenberg matrix output from ELMHES. ELTRAN-S Accumulates the stabilized elementary similarity transformations used in the reduction of a real general matrix to upper Hessenberg form by ELMHES. HTRIB3-S Compute the eigenvectors of a complex Hermitian matrix from the eigenvectors of a real symmetric tridiagonal matrix output from HTRID3. HTRIBK-S Form the eigenvectors of a complex Hermitian matrix from the eigenvectors of a real symmetric tridiagonal matrix output from HTRIDI. ORTBAK-S Form the eigenvectors of a general real matrix from the CORTB-C eigenvectors of the upper Hessenberg matrix output from ORTHES. ORTRAN-S Accumulate orthogonal similarity transformations in the reduction of real general matrix by ORTHES. REBAK-S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC or REDUC2. REBAKB-S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC2. TRBAK1-S Form the eigenvectors of real symmetric matrix from the eigenvectors of a symmetric tridiagonal matrix formed by TRED1. TRBAK3-S Form the eigenvectors of a real symmetric matrix from the eigenvectors of a symmetric tridiagonal matrix formed by TRED3. D5. QR decomposition, Gram-Schmidt orthogonalization LLSIA-S Solve a linear least squares problems by performing a QR DLLSIA-D factorization of the matrix using Householder transformations. Emphasis is put on detecting possible rank deficiency. SGLSS-S Solve a linear least squares problems by performing a QR DGLSS-D factorization of the matrix using Householder transformations. Emphasis is put on detecting possible rank deficiency. SQRDC-S Use Householder transformations to compute the QR DQRDC-D factorization of an N by P matrix. Column pivoting is a CQRDC-C users option. D6. Singular value decomposition SSVDC-S Perform the singular value decomposition of a rectangular DSVDC-D matrix. CSVDC-C D7. Update matrix decompositions D7B. Cholesky SCHDD-S Downdate an augmented Cholesky decomposition or the DCHDD-D triangular factor of an augmented QR decomposition. CCHDD-C SCHEX-S Update the Cholesky factorization A=TRANS(R)*R of A DCHEX-D positive definite matrix A of order P under diagonal CCHEX-C permutations of the form TRANS(E)*A*E, where E is a permutation matrix. SCHUD-S Update an augmented Cholesky decomposition of the DCHUD-D triangular part of an augmented QR decomposition. CCHUD-C D9. Overdetermined or underdetermined systems of equations, singular systems, pseudo-inverses (search also classes D5, D6, K1a, L8a) BNDACC-S Compute the LU factorization of a banded matrices using DBNDAC-D sequential accumulation of rows of the data matrix. Exactly one right-hand side vector is permitted. BNDSOL-S Solve the least squares problem for a banded matrix using DBNDSL-D sequential accumulation of rows of the data matrix. Exactly one right-hand side vector is permitted. HFTI-S Solve a linear least squares problems by performing a QR DHFTI-D factorization of the matrix using Householder transformations. LLSIA-S Solve a linear least squares problems by performing a QR DLLSIA-D factorization of the matrix using Householder transformations. Emphasis is put on detecting possible rank deficiency. LSEI-S Solve a linearly constrained least squares problem with DLSEI-D equality and inequality constraints, and optionally compute a covariance matrix. MINFIT-S Compute the singular value decomposition of a rectangular matrix and solve the related linear least squares problem. SGLSS-S Solve a linear least squares problems by performing a QR DGLSS-D factorization of the matrix using Householder transformations. Emphasis is put on detecting possible rank deficiency. SQRSL-S Apply the output of SQRDC to compute coordinate transfor- DQRSL-D mations, projections, and least squares solutions. CQRSL-C ULSIA-S Solve an underdetermined linear system of equations by DULSIA-D performing an LQ factorization of the matrix using Householder transformations. Emphasis is put on detecting possible rank deficiency.