# complex16GTcomputational - Man Page

complex16

## Synopsis

### Functions

subroutine **zgtcon** (NORM, **N**, DL, D, DU, DU2, IPIV, ANORM, RCOND, WORK, INFO)**ZGTCON**

subroutine **zgtrfs** (TRANS, **N**, **NRHS**, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, **LDB**, X, LDX, FERR, BERR, WORK, RWORK, INFO)**ZGTRFS**

subroutine **zgttrf** (**N**, DL, D, DU, DU2, IPIV, INFO)**ZGTTRF**

subroutine **zgttrs** (TRANS, **N**, **NRHS**, DL, D, DU, DU2, IPIV, B, **LDB**, INFO)**ZGTTRS**

subroutine **zgtts2** (ITRANS, **N**, **NRHS**, DL, D, DU, DU2, IPIV, B, **LDB**)**ZGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

## Detailed Description

This is the group of complex16 computational functions for GT matrices

## Function Documentation

### subroutine zgtcon (character NORM, integer N, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, double precision ANORM, double precision RCOND, complex*16, dimension( * ) WORK, integer INFO)

**ZGTCON**

**Purpose:**

ZGTCON estimates the reciprocal of the condition number of a complex tridiagonal matrix A using the LU factorization as computed by ZGTTRF. 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*16 array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by ZGTTRF.

*D*D is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.

*DU*DU is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.

*DU2*DU2 is COMPLEX*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 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.

Definition at line 139 of file zgtcon.f.

### subroutine zgtrfs (character TRANS, integer N, integer NRHS, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DLF, complex*16, dimension( * ) DF, complex*16, dimension( * ) DUF, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, complex*16, dimension( * ) WORK, double precision, dimension( * ) RWORK, integer INFO)

**ZGTRFS**

**Purpose:**

ZGTRFS 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*16 array, dimension (N-1) The (n-1) subdiagonal elements of A.

*D*D is COMPLEX*16 array, dimension (N) The diagonal elements of A.

*DU*DU is COMPLEX*16 array, dimension (N-1) The (n-1) superdiagonal elements of A.

*DLF*DLF is COMPLEX*16 array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by ZGTTRF.

*DF*DF is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.

*DUF*DUF is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first superdiagonal of U.

*DU2*DU2 is COMPLEX*16 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*16 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*16 array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by ZGTTRS. 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (2*N)

*RWORK*RWORK is DOUBLE PRECISION 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.

Definition at line 207 of file zgtrfs.f.

### subroutine zgttrf (integer N, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, integer INFO)

**ZGTTRF**

**Purpose:**

ZGTTRF 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*16 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*16 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*16 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*16 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.

Definition at line 123 of file zgttrf.f.

### subroutine zgttrs (character TRANS, integer N, integer NRHS, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB, integer INFO)

**ZGTTRS**

**Purpose:**

ZGTTRS 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 ZGTTRF.

**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*16 array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A.

*D*D is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.

*DU*DU is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first super-diagonal of U.

*DU2*DU2 is COMPLEX*16 array, dimension (N-2) The (n-2) elements of the second super-diagonal of U.

*IPIV**B*B is COMPLEX*16 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.

Definition at line 136 of file zgttrs.f.

### subroutine zgtts2 (integer ITRANS, integer N, integer NRHS, complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * ) DU, complex*16, dimension( * ) DU2, integer, dimension( * ) IPIV, complex*16, dimension( ldb, * ) B, integer LDB)

**ZGTTS2** solves a system of linear equations with a tridiagonal matrix using the LU factorization computed by sgttrf.

**Purpose:**

ZGTTS2 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 ZGTTRF.

**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*16 array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A.

*D*D is COMPLEX*16 array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A.

*DU*DU is COMPLEX*16 array, dimension (N-1) The (n-1) elements of the first super-diagonal of U.

*DU2*DU2 is COMPLEX*16 array, dimension (N-2) The (n-2) elements of the second super-diagonal of U.

*IPIV**B*B is COMPLEX*16 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.

Definition at line 127 of file zgtts2.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man pages zgtcon(3), zgtrfs(3), zgttrf(3), zgttrs(3) and zgtts2(3) are aliases of complex16GTcomputational(3).