--- name: "complexPTsolve" section: "3" platform: "linux" package: "liblapack-doc" version: "3.11.0-2" release: "Debian 12 (Bookworm)" platforms: ["linux"] url: "https://new.carta.tech/complexPTsolve" --- # complexPTsolve(3) complex ## NAME complexPTsolve - complex ## SYNOPSIS ### Functions subroutine **cptsv** (N, NRHS, D, E, B, LDB, INFO)\ **CPTSV computes the solution to system of linear equations A \* X = B for PT matrices** subroutine **cptsvx** (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)\ **CPTSVX computes the solution to system of linear equations A \* X = B for PT matrices** ## Detailed Description This is the group of complex solve driver functions for PT matrices ## Function Documentation ### subroutine cptsv (integer N, integer NRHS, real, dimension( \* ) D, complex, dimension( \* ) E, complex, dimension( ldb, \* ) B, integer LDB, integer INFO) **CPTSV computes the solution to system of linear equations A \* X = B for PT matrices** **Purpose:** ``` CPTSV computes the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix, and X and B are N-by-NRHS matrices. A is factored as A = L*D*L**H, and the factored form of A is then used to solve the system of equations. ``` **Parameters** *N* ``` N is INTEGER The order of the matrix A. N >= 0. ``` *NRHS* ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. ``` *D* ``` D is REAL array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, the n diagonal elements of the diagonal matrix D from the factorization A = L*D*L**H. ``` *E* ``` E is COMPLEX array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A. On exit, the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. E can also be regarded as the superdiagonal of the unit bidiagonal factor U from the U**H*D*U factorization of A. ``` *B* ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if INFO = 0, the N-by-NRHS solution matrix X. ``` *LDB* ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). ``` *INFO* ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the leading minor of order i is not positive definite, and the solution has not been computed. The factorization has not been completed unless i = N. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine cptsvx (character FACT, integer N, integer NRHS, real, dimension( \* ) D, complex, dimension( \* ) E, real, dimension( \* ) DF, complex, dimension( \* ) EF, complex, dimension( ldb, \* ) B, integer LDB, complex, dimension( ldx, \* ) X, integer LDX, real RCOND, real, dimension( \* ) FERR, real, dimension( \* ) BERR, complex, dimension( \* ) WORK, real, dimension( \* ) RWORK, integer INFO) **CPTSVX computes the solution to system of linear equations A \* X = B for PT matrices** **Purpose:** ``` CPTSVX uses the factorization A = L*D*L**H to compute the solution to a complex system of linear equations A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal matrix and X and B are N-by-NRHS matrices. Error bounds on the solution and a condition estimate are also provided. ``` **Description:** ``` The following steps are performed: 1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U**H*D*U. 2. If the leading i-by-i principal minor is not positive definite, then the routine returns with INFO = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. 3. The system of equations is solved for X using the factored form of A. 4. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. ``` **Parameters** *FACT* ``` FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry. = 'F': On entry, DF and EF contain the factored form of A. D, E, DF, and EF will not be modified. = 'N': The matrix A will be copied to DF and EF and factored. ``` *N* ``` N is INTEGER The order of the matrix A. N >= 0. ``` *NRHS* ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. ``` *D* ``` D is REAL array, dimension (N) The n diagonal elements of the tridiagonal matrix A. ``` *E* ``` E is COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of the tridiagonal matrix A. ``` *DF* ``` DF is REAL array, dimension (N) If FACT = 'F', then DF is an input argument and on entry contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. If FACT = 'N', then DF is an output argument and on exit contains the n diagonal elements of the diagonal matrix D from the L*D*L**H factorization of A. ``` *EF* ``` EF is COMPLEX array, dimension (N-1) If FACT = 'F', then EF is an input argument and on entry contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. If FACT = 'N', then EF is an output argument and on exit contains the (n-1) subdiagonal elements of the unit bidiagonal factor L from the L*D*L**H factorization of A. ``` *B* ``` B is COMPLEX array, dimension (LDB,NRHS) The N-by-NRHS right hand side matrix B. ``` *LDB* ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). ``` *X* ``` X is COMPLEX array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X. ``` *LDX* ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). ``` *RCOND* ``` RCOND is REAL The reciprocal condition number of the matrix A. If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. ``` *FERR* ``` FERR is REAL array, dimension (NRHS) The forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution corresponding to X(j), FERR(j) is an estimated upper bound for the magnitude of the largest element in (X(j) - XTRUE) divided by the magnitude of the largest element in X(j). ``` *BERR* ``` BERR is REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). ``` *WORK* ``` WORK is COMPLEX array, dimension (N) ``` *RWORK* ``` RWORK is REAL array, dimension (N) ``` *INFO* ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ## Author Generated automatically by Doxygen for LAPACK from the source code.