--- name: "complexGTcomputational" section: "3" platform: "linux" package: "liblapack-doc" version: "3.11.0-2" release: "Debian 12 (Bookworm)" platforms: ["linux"] url: "https://new.carta.tech/complexGTcomputational" --- # complexGTcomputational(3) complex ## NAME complexGTcomputational - complex ## SYNOPSIS ### Functions subroutine **cgtcon** (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)\ **CGTCON** subroutine **cgtrfs** (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)\ **CGTRFS** subroutine **cgttrf** (N, DL, D, DU, DU2, IPIV, INFO)\ **CGTTRF** subroutine **cgttrs** (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)\ **CGTTRS** subroutine **cgtts2** (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)\ **CGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf. ## Detailed Description This is the group of complex computational functions for GT matrices ## Function Documentation ### subroutine cgtcon (character NORM, integer N, complex, dimension( \* ) DL, complex, dimension( \* ) D, complex, dimension( \* ) DU, complex, dimension( \* ) DU2, integer, dimension( \* ) IPIV, real ANORM, real RCOND, complex, dimension( \* ) WORK, integer INFO) **CGTCON** **Purpose:** ``` CGTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by CGTTRF. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). ``` **Parameters** *NORM* ``` NORM is CHARACTER*1 Specifies whether the 1-norm condition number or the infinity-norm condition number is required: = '1' or 'O': 1-norm; = 'I': Infinity-norm. ``` *N* ``` N is INTEGER The order of the matrix A. N >= 0. ``` *DL* ``` DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by CGTTRF. ``` *D* ``` D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. ``` *DU2* ``` DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second superdiagonal of U. ``` *IPIV* ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. ``` *ANORM* ``` ANORM is REAL If NORM = '1' or 'O', the 1-norm of the original matrix A. If NORM = 'I', the infinity-norm of the original matrix A. ``` *RCOND* ``` RCOND is REAL The reciprocal of the condition number of the matrix A, computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an estimate of the 1-norm of inv(A) computed in this routine. ``` *WORK* ``` WORK is COMPLEX array, dimension (2*N) ``` *INFO* ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine cgtrfs (character TRANS, integer N, integer NRHS, complex, dimension( \* ) DL, complex, dimension( \* ) D, complex, dimension( \* ) DU, complex, dimension( \* ) DLF, complex, dimension( \* ) DF, complex, dimension( \* ) DUF, complex, dimension( \* ) DU2, integer, dimension( \* ) IPIV, complex, dimension( ldb, \* ) B, integer LDB, complex, dimension( ldx, \* ) X, integer LDX, real, dimension( \* ) FERR, real, dimension( \* ) BERR, complex, dimension( \* ) WORK, real, dimension( \* ) RWORK, integer INFO) **CGTRFS** **Purpose:** ``` CGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. ``` **Parameters** *TRANS* ``` TRANS is CHARACTER*1 Specifies the form of the system of equations: = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) ``` *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. ``` *DL* ``` DL is COMPLEX array, dimension (N-1) The (n-1) subdiagonal elements of A. ``` *D* ``` D is COMPLEX array, dimension (N) The diagonal elements of A. ``` *DU* ``` DU is COMPLEX array, dimension (N-1) The (n-1) superdiagonal elements of A. ``` *DLF* ``` DLF is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by CGTTRF. ``` *DF* ``` DF is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DUF* ``` DUF is COMPLEX array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. ``` *DU2* ``` DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second superdiagonal of U. ``` *IPIV* ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. ``` *B* ``` B is COMPLEX array, dimension (LDB,NRHS) The 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) On entry, the solution matrix X, as computed by CGTTRS. On exit, the improved solution matrix X. ``` *LDX* ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N). ``` *FERR* ``` FERR is REAL array, dimension (NRHS) The estimated 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). The estimate is as reliable as the estimate for RCOND, and is almost always a slight overestimate of the true error. ``` *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 (2*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 ``` **Internal Parameters:** ``` ITMAX is the maximum number of steps of iterative refinement. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine cgttrf (integer N, complex, dimension( \* ) DL, complex, dimension( \* ) D, complex, dimension( \* ) DU, complex, dimension( \* ) DU2, integer, dimension( \* ) IPIV, integer INFO) **CGTTRF** **Purpose:** ``` CGTTRF computes an LU factorization of a complex tridiagonal matrix A using elimination with partial pivoting and row interchanges. The factorization has the form A = L * U where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two superdiagonals. ``` **Parameters** *N* ``` N is INTEGER The order of the matrix A. ``` *DL* ``` DL is COMPLEX array, dimension (N-1) On entry, DL must contain the (n-1) sub-diagonal elements of A. On exit, DL is overwritten by the (n-1) multipliers that define the matrix L from the LU factorization of A. ``` *D* ``` D is COMPLEX array, dimension (N) On entry, D must contain the diagonal elements of A. On exit, D is overwritten by the n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is COMPLEX array, dimension (N-1) On entry, DU must contain the (n-1) super-diagonal elements of A. On exit, DU is overwritten by the (n-1) elements of the first super-diagonal of U. ``` *DU2* ``` DU2 is COMPLEX array, dimension (N-2) On exit, DU2 is overwritten by the (n-2) elements of the second super-diagonal of U. ``` *IPIV* ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. ``` *INFO* ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value > 0: if INFO = k, U(k,k) is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations. ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine cgttrs (character TRANS, integer N, integer NRHS, complex, dimension( \* ) DL, complex, dimension( \* ) D, complex, dimension( \* ) DU, complex, dimension( \* ) DU2, integer, dimension( \* ) IPIV, complex, dimension( ldb, \* ) B, integer LDB, integer INFO) **CGTTRS** **Purpose:** ``` CGTTRS solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF. ``` **Parameters** *TRANS* ``` TRANS is CHARACTER*1 Specifies the form of the system of equations. = 'N': A * X = B (No transpose) = 'T': A**T * X = B (Transpose) = 'C': A**H * X = B (Conjugate transpose) ``` *N* ``` N is INTEGER The order of the matrix A. ``` *NRHS* ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. ``` *DL* ``` DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. ``` *D* ``` D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first super-diagonal of U. ``` *DU2* ``` DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second super-diagonal of U. ``` *IPIV* ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. ``` *B* ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors 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 = -k, the k-th argument had an illegal value ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine cgtts2 (integer ITRANS, integer N, integer NRHS, complex, dimension( \* ) DL, complex, dimension( \* ) D, complex, dimension( \* ) DU, complex, dimension( \* ) DU2, integer, dimension( \* ) IPIV, complex, dimension( ldb, \* ) B, integer LDB) **CGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf. **Purpose:** ``` CGTTS2 solves one of the systems of equations A * X = B, A**T * X = B, or A**H * X = B, with a tridiagonal matrix A using the LU factorization computed by CGTTRF. ``` **Parameters** *ITRANS* ``` ITRANS is INTEGER Specifies the form of the system of equations. = 0: A * X = B (No transpose) = 1: A**T * X = B (Transpose) = 2: A**H * X = B (Conjugate transpose) ``` *N* ``` N is INTEGER The order of the matrix A. ``` *NRHS* ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0. ``` *DL* ``` DL is COMPLEX array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. ``` *D* ``` D is COMPLEX array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is COMPLEX array, dimension (N-1) The (n-1) elements of the first super-diagonal of U. ``` *DU2* ``` DU2 is COMPLEX array, dimension (N-2) The (n-2) elements of the second super-diagonal of U. ``` *IPIV* ``` IPIV is INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. ``` *B* ``` B is COMPLEX array, dimension (LDB,NRHS) On entry, the matrix of right hand side vectors B. On exit, B is overwritten by the solution vectors X. ``` *LDB* ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N). ``` **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.