# ptrfs - Man Page

ptrfs: iterative refinement

## Synopsis

### Functions

subroutine cptrfs (uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CPTRFS
subroutine dptrfs (n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
DPTRFS
subroutine sptrfs (n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
SPTRFS
subroutine zptrfs (uplo, n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, rwork, info)
ZPTRFS

## Function Documentation

### subroutine cptrfs (character uplo, 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, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CPTRFS

Purpose:

``` CPTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite
and tridiagonal, and provides error bounds and backward error
estimates for the solution.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the superdiagonal or the subdiagonal of the
tridiagonal matrix A is stored and the form of the
factorization:
= 'U':  E is the superdiagonal of A, and A = U**H*D*U;
= 'L':  E is the subdiagonal of A, and A = L*D*L**H.
(The two forms are equivalent if A is real.)```

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)
The n real diagonal elements of the tridiagonal matrix A.```

E

```          E is COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix A
(see UPLO).```

DF

```          DF is REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from
the factorization computed by CPTTRF.```

EF

```          EF is COMPLEX array, dimension (N-1)
The (n-1) off-diagonal elements of the unit bidiagonal
factor U or L from the factorization computed by CPTTRF
(see UPLO).```

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 CPTTRS.
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 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```

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 181 of file cptrfs.f.

### subroutine dptrfs (integer n, integer nrhs, double precision, dimension( * ) d, double precision, dimension( * ) e, double precision, dimension( * ) df, double precision, dimension( * ) ef, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer info)

DPTRFS

Purpose:

``` DPTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and tridiagonal, and provides error bounds and backward error
estimates for the solution.```
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 DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.```

E

```          E is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.```

DF

```          DF is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
factorization computed by DPTTRF.```

EF

```          EF is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal factor
L from the factorization computed by DPTTRF.```

B

```          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the solution matrix X, as computed by DPTTRS.
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 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 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 DOUBLE PRECISION array, dimension (2*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 161 of file dptrfs.f.

### subroutine sptrfs (integer n, integer nrhs, real, dimension( * ) d, real, dimension( * ) e, real, dimension( * ) df, real, dimension( * ) ef, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldx, * ) x, integer ldx, real, dimension( * ) ferr, real, dimension( * ) berr, real, dimension( * ) work, integer info)

SPTRFS

Purpose:

``` SPTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive definite
and tridiagonal, and provides error bounds and backward error
estimates for the solution.```
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)
The n diagonal elements of the tridiagonal matrix A.```

E

```          E is REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.```

DF

```          DF is REAL array, dimension (N)
The n diagonal elements of the diagonal matrix D from the
factorization computed by SPTTRF.```

EF

```          EF is REAL array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal factor
L from the factorization computed by SPTTRF.```

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 SPTTRS.
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 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 REAL array, dimension (2*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 161 of file sptrfs.f.

### subroutine zptrfs (character uplo, integer n, integer nrhs, double precision, dimension( * ) d, complex*16, dimension( * ) e, double precision, dimension( * ) df, complex*16, dimension( * ) ef, 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)

ZPTRFS

Purpose:

``` ZPTRFS improves the computed solution to a system of linear
equations when the coefficient matrix is Hermitian positive definite
and tridiagonal, and provides error bounds and backward error
estimates for the solution.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the superdiagonal or the subdiagonal of the
tridiagonal matrix A is stored and the form of the
factorization:
= 'U':  E is the superdiagonal of A, and A = U**H*D*U;
= 'L':  E is the subdiagonal of A, and A = L*D*L**H.
(The two forms are equivalent if A is real.)```

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 DOUBLE PRECISION array, dimension (N)
The n real diagonal elements of the tridiagonal matrix A.```

E

```          E is COMPLEX*16 array, dimension (N-1)
The (n-1) off-diagonal elements of the tridiagonal matrix A
(see UPLO).```

DF

```          DF is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D from
the factorization computed by ZPTTRF.```

EF

```          EF is COMPLEX*16 array, dimension (N-1)
The (n-1) off-diagonal elements of the unit bidiagonal
factor U or L from the factorization computed by ZPTTRF
(see UPLO).```

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 ZPTTRS.
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 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 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 (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 181 of file zptrfs.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK