## table of contents

complex16GTcomputational(3) | LAPACK | complex16GTcomputational(3) |

# NAME¶

complex16GTcomputational - 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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*

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)

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

## 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*

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)

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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

# Author¶

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