realPTsolve(3) LAPACK realPTsolve(3)

realPTsolve

# SYNOPSIS¶

## Functions¶

subroutine sptsv (N, NRHS, D, E, B, LDB, INFO)
SPTSV computes the solution to system of linear equations A * X = B for PT matrices subroutine sptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, INFO)
SPTSVX computes the solution to system of linear equations A * X = B for PT matrices

# Detailed Description¶

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

# Function Documentation¶

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

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

Purpose:

```
SPTSV computes the solution to a real system of linear equations

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

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

A is factored as A = L*D*L**T, 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**T.```

E

```
E is REAL 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**T factorization of

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

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

B

```
B is REAL 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 sptsvx (character FACT, 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 RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer INFO)¶

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

Purpose:

```
SPTSVX uses the factorization A = L*D*L**T to compute the solution

to a real system of linear equations A*X = B, where A is an N-by-N

symmetric 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**T, where L

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

factorization can also be regarded as having the form

A = U**T*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 A has been

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 REAL 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**T 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**T factorization of A.```

EF

```
EF is REAL 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**T 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**T factorization of A.```

B

```
B is REAL 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 REAL array, dimension (LDX,NRHS)

If INFO = 0 of 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 REAL array, dimension (2*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