--- name: "realGTcomputational" section: "3" platform: "linux" package: "liblapack-doc" version: "3.11.0-2" release: "Debian 12 (Bookworm)" platforms: ["linux"] url: "https://new.carta.tech/realGTcomputational" --- # realGTcomputational(3) real ## NAME realGTcomputational - real ## SYNOPSIS ### Functions subroutine **sgtcon** (NORM, N, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, IWORK, INFO)\ **SGTCON** subroutine **sgtrfs** (TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)\ **SGTRFS** subroutine **sgttrf** (N, DL, D, DU, DU2, IPIV, INFO)\ **SGTTRF** subroutine **sgttrs** (TRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, INFO)\ **SGTTRS** subroutine **sgtts2** (ITRANS, N, NRHS, DL, D, DU, DU2, IPIV, B, LDB)\ **SGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf. ## Detailed Description This is the group of real computational functions for GT matrices ## Function Documentation ### subroutine sgtcon (character NORM, integer N, real, dimension( \* ) DL, real, dimension( \* ) D, real, dimension( \* ) DU, real, dimension( \* ) DU2, integer, dimension( \* ) IPIV, real ANORM, real RCOND, real, dimension( \* ) WORK, integer, dimension( \* ) IWORK, integer INFO) **SGTCON** **Purpose:** ``` SGTCON estimates the reciprocal of the condition number of a real tridiagonal matrix A using the LU factorization as computed by SGTTRF. 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 REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF. ``` *D* ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. ``` *DU2* ``` DU2 is REAL 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 REAL array, dimension (2*N) ``` *IWORK* ``` IWORK is INTEGER array, dimension (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 sgtrfs (character TRANS, integer N, integer NRHS, real, dimension( \* ) DL, real, dimension( \* ) D, real, dimension( \* ) DU, real, dimension( \* ) DLF, real, dimension( \* ) DF, real, dimension( \* ) DUF, real, dimension( \* ) DU2, integer, dimension( \* ) IPIV, real, dimension( ldb, \* ) B, integer LDB, real, dimension( ldx, \* ) X, integer LDX, real, dimension( \* ) FERR, real, dimension( \* ) BERR, real, dimension( \* ) WORK, integer, dimension( \* ) IWORK, integer INFO) **SGTRFS** **Purpose:** ``` SGTRFS 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 = 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 REAL array, dimension (N-1) The (n-1) subdiagonal elements of A. ``` *D* ``` D is REAL array, dimension (N) The diagonal elements of A. ``` *DU* ``` DU is REAL array, dimension (N-1) The (n-1) superdiagonal elements of A. ``` *DLF* ``` DLF is REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by SGTTRF. ``` *DF* ``` DF is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DUF* ``` DUF is REAL array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. ``` *DU2* ``` DU2 is REAL 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 REAL 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 REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGTTRS. 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 REAL array, dimension (3*N) ``` *IWORK* ``` IWORK is INTEGER 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 sgttrf (integer N, real, dimension( \* ) DL, real, dimension( \* ) D, real, dimension( \* ) DU, real, dimension( \* ) DU2, integer, dimension( \* ) IPIV, integer INFO) **SGTTRF** **Purpose:** ``` SGTTRF computes an LU factorization of a real 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 REAL 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 REAL 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 REAL 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 REAL 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 sgttrs (character TRANS, integer N, integer NRHS, real, dimension( \* ) DL, real, dimension( \* ) D, real, dimension( \* ) DU, real, dimension( \* ) DU2, integer, dimension( \* ) IPIV, real, dimension( ldb, \* ) B, integer LDB, integer INFO) **SGTTRS** **Purpose:** ``` SGTTRS solves one of the systems of equations A*X = B or A**T*X = B, with a tridiagonal matrix A using the LU factorization computed by SGTTRF. ``` **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**T* X = B (Conjugate transpose = 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 REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. ``` *D* ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first super-diagonal of U. ``` *DU2* ``` DU2 is REAL 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 REAL 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 = -i, the i-th argument had an illegal value ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. ### subroutine sgtts2 (integer ITRANS, integer N, integer NRHS, real, dimension( \* ) DL, real, dimension( \* ) D, real, dimension( \* ) DU, real, dimension( \* ) DU2, integer, dimension( \* ) IPIV, real, dimension( ldb, \* ) B, integer LDB) **SGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf. **Purpose:** ``` SGTTS2 solves one of the systems of equations A*X = B or A**T*X = B, with a tridiagonal matrix A using the LU factorization computed by SGTTRF. ``` **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**T* X = B (Conjugate transpose = 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 REAL array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A. ``` *D* ``` D is REAL array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. ``` *DU* ``` DU is REAL array, dimension (N-1) The (n-1) elements of the first super-diagonal of U. ``` *DU2* ``` DU2 is REAL 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 REAL 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.