doubleGTsolve(3) LAPACK doubleGTsolve(3)

doubleGTsolve

# SYNOPSIS¶

## Functions¶

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

# Detailed Description¶

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

# Function Documentation¶

## subroutine dgtsv (integer N, integer NRHS, double precision, dimension( * ) DL, double precision, dimension( * ) D, double precision, dimension( * ) DU, double precision, dimension( ldb, * ) B, integer LDB, integer INFO)¶

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

Purpose:

```
DGTSV  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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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

NAG Ltd.

Date

December 2016

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

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

Purpose:

```
DGTSVX 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 DOUBLE PRECISION array, dimension (N-1)

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

D

```
D is DOUBLE PRECISION array, dimension (N)

The n diagonal elements of A.```

DU

```
DU is DOUBLE PRECISION array, dimension (N-1)

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

DLF

```
DLF is DOUBLE PRECISION 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 DGTTRF.

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DGTTRF.

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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION

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 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 DOUBLE PRECISION 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