complexPTsolve(3) LAPACK complexPTsolve(3)

complexPTsolve

# SYNOPSIS¶

## Functions¶

subroutine cptsv (N, NRHS, D, E, B, LDB, INFO)
CPTSV computes the solution to system of linear equations A * X = B for PT matrices subroutine cptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CPTSVX computes the solution to system of linear equations A * X = B for PT matrices

# Detailed Description¶

This is the group of complex solve driver functions for PT matrices

# Function Documentation¶

## subroutine cptsv (integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB, integer INFO)¶

CPTSV computes the solution to system of linear equations A * X = B for PT matrices

Purpose:

```
CPTSV computes the solution to a complex system of linear equations

A*X = B, where A is an N-by-N Hermitian positive definite tridiagonal

matrix, and X and B are N-by-NRHS matrices.

A is factored as A = L*D*L**H, and the factored form of A is then

used to solve the system of equations.```

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)

On entry, the n diagonal elements of the tridiagonal matrix

A.  On exit, the n diagonal elements of the diagonal matrix

D from the factorization A = L*D*L**H.```

E

```
E is COMPLEX array, dimension (N-1)

On entry, the (n-1) subdiagonal elements of the tridiagonal

matrix A.  On exit, the (n-1) subdiagonal elements of the

unit bidiagonal factor L from the L*D*L**H factorization of

A.  E can also be regarded as the superdiagonal of the unit

bidiagonal factor U from the U**H*D*U factorization of A.```

B

```
B is COMPLEX array, dimension (LDB,NRHS)

On entry, the N-by-NRHS right hand side matrix B.

On exit, if INFO = 0, the N-by-NRHS solution matrix 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

> 0:  if INFO = i, the leading minor of order i is not

positive definite, and the solution has not been

computed.  The factorization has not been completed

unless i = N.```

Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Date

December 2016

## subroutine cptsvx (character FACT, 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 RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO)¶

CPTSVX computes the solution to system of linear equations A * X = B for PT matrices

Purpose:

```
CPTSVX uses the factorization A = L*D*L**H to compute the solution

to a complex system of linear equations A*X = B, where A is an

N-by-N Hermitian positive definite tridiagonal matrix and X and B

are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also

provided.```

Description:

```
The following steps are performed:

1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L

is a unit lower bidiagonal matrix and D is diagonal.  The

factorization can also be regarded as having the form

A = U**H*D*U.

2. If the leading i-by-i principal minor is not positive definite,

then the routine returns with INFO = i. Otherwise, the factored

form of A is used to estimate the condition number of the matrix

A.  If the reciprocal of the condition number is less than machine

precision, INFO = N+1 is returned as a warning, but the routine

still goes on to solve for X and compute error bounds as

described below.

3. The system of equations is solved for X using the factored form

of A.

4. Iterative refinement is applied to improve the computed solution

matrix and calculate error bounds and backward error estimates

for it.```

Parameters

FACT

```
FACT is CHARACTER*1

Specifies whether or not the factored form of the matrix

A is supplied on entry.

= 'F':  On entry, DF and EF contain the factored form of A.

D, E, DF, and EF will not be modified.

= 'N':  The matrix A will be copied to DF and EF and

factored.```

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 matrices B and X.  NRHS >= 0.```

D

```
D is REAL array, dimension (N)

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

E

```
E is COMPLEX array, dimension (N-1)

The (n-1) subdiagonal elements of the tridiagonal matrix A.```

DF

```
DF is REAL array, dimension (N)

If FACT = 'F', then DF is an input argument and on entry

contains the n diagonal elements of the diagonal matrix D

from the L*D*L**H factorization of A.

If FACT = 'N', then DF is an output argument and on exit

contains the n diagonal elements of the diagonal matrix D

from the L*D*L**H factorization of A.```

EF

```
EF is COMPLEX array, dimension (N-1)

If FACT = 'F', then EF is an input argument and on entry

contains the (n-1) subdiagonal elements of the unit

bidiagonal factor L from the L*D*L**H factorization of A.

If FACT = 'N', then EF is an output argument and on exit

contains the (n-1) subdiagonal elements of the unit

bidiagonal factor L from the L*D*L**H factorization of A.```

B

```
B is COMPLEX array, dimension (LDB,NRHS)

The N-by-NRHS 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)

If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.```

LDX

```
LDX is INTEGER

The leading dimension of the array X.  LDX >= max(1,N).```

RCOND

```
RCOND is REAL

The reciprocal condition number of the matrix A.  If RCOND

is less than the machine precision (in particular, if

RCOND = 0), the matrix is singular to working precision.

This condition is indicated by a return code of INFO > 0.```

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

> 0:  if INFO = i, and i is

<= N:  the leading minor of order i of A is

not positive definite, so the factorization

could not be completed, and the solution has not

been computed. RCOND = 0 is returned.

= N+1: U is nonsingular, but RCOND is less than machine

precision, meaning that the matrix is singular

to working precision.  Nevertheless, the

solution and error bounds are computed because

there are a number of situations where the

computed solution can be more accurate than the

value of RCOND would suggest.```

Author

Univ. of Tennessee

Univ. of California Berkeley