realGTsolve(3) LAPACK realGTsolve(3)

realGTsolve

SYNOPSIS¶

Functions¶

subroutine sgtsv (N, NRHS, DL, D, DU, B, LDB, INFO)
SGTSV computes the solution to system of linear equations A * X = B for GT matrices subroutine sgtsvx (FACT, TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Detailed Description¶

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

Function Documentation¶

subroutine sgtsv (integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( ldb, * ) B, integer LDB, integer INFO)¶

SGTSV computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

```
SGTSV  solves the equation

A*X = B,

where A is an n by n tridiagonal matrix, by Gaussian elimination with

partial pivoting.

Note that the equation  A**T*X = B  may be solved by interchanging the

order of the arguments DU and DL.```

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.```

DL

```
DL is REAL 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-2) elements of the

second super-diagonal of the upper triangular matrix U from

the LU factorization of A, in DL(1), ..., DL(n-2).```

D

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

On entry, D must contain the diagonal elements of A.

On exit, D is overwritten by the n diagonal elements of U.```

DU

```
DU is REAL 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.```

B

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

On entry, the N by NRHS matrix of 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, U(i,i) is exactly zero, and the solution

has not been computed.  The factorization has not been

completed unless i = N.```

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

subroutine sgtsvx (character FACT, character TRANS, integer N, integer NRHS, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( * ) DLF, real, dimension( * ) DF, real, dimension( * ) DUF, real, dimension( * ) DU2, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)¶

SGTSVX computes the solution to system of linear equations A * X = B for GT matrices

Purpose:

```
SGTSVX uses the LU factorization to compute the solution to a real

system of linear equations A * X = B or A**T * X = B,

where A is a tridiagonal matrix of order N 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 LU decomposition is used to factor the matrix A

as 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.

2. If some U(i,i)=0, so that U is exactly singular, 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':  DLF, DF, DUF, DU2, and IPIV contain the factored

form of A; DL, D, DU, DLF, DF, DUF, DU2 and IPIV

will not be modified.

= 'N':  The matrix will be copied to DLF, DF, and DUF

and factored.```

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 = 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 REAL array, dimension (N-1)

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

D

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

The n diagonal elements of A.```

DU

```
DU is REAL array, dimension (N-1)

The (n-1) superdiagonal elements of A.```

DLF

```
DLF is REAL array, dimension (N-1)

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

contains the (n-1) multipliers that define the matrix L from

the LU factorization of A as computed by SGTTRF.

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

contains the (n-1) multipliers that define the matrix L from

the LU factorization of 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 upper triangular

matrix U from the LU factorization of A.

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

contains the n diagonal elements of the upper triangular

matrix U from the LU factorization of A.```

DUF

```
DUF is REAL array, dimension (N-1)

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

contains the (n-1) elements of the first superdiagonal of U.

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

contains the (n-1) elements of the first superdiagonal of U.```

DU2

```
DU2 is REAL array, dimension (N-2)

If FACT = 'F', then DU2 is an input argument and on entry

contains the (n-2) elements of the second superdiagonal of

U.

If FACT = 'N', then DU2 is an output argument and on exit

contains the (n-2) elements of the second superdiagonal of

U.```

IPIV

```
IPIV is INTEGER array, dimension (N)

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

contains the pivot indices from the LU factorization of A as

computed by SGTTRF.

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

contains the pivot indices from the LU factorization of A;

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 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 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 estimate of 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 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 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 (3*N)```

IWORK

```
IWORK is INTEGER 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:  U(i,i) is exactly zero.  The factorization

has not been completed unless i = N, but the

factor U is exactly singular, so the solution

and error bounds could not be 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

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

Author¶

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