?gtsvx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/22/2024

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?gtsvx

Computes the solution to the real or complex system of linear equations with a tridiagonal coefficient matrix A and multiple right-hand sides, and provides error bounds on the solution.

Syntax

call sgtsvx( fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call dgtsvx( fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info )

call cgtsvx( fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call zgtsvx( fact, trans, n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info )

call gtsvx( dl, d, du, b, x [,dlf] [,df] [,duf] [,du2] [,ipiv] [,fact] [,trans] [,ferr] [,berr] [,rcond] [,info] )

Include Files

mkl.fi, lapack.f90

Description

The routine uses the LU factorization to compute the solution to a real or complex system of linear equations A*X = B, A^T*X = B, or A^H*X = B, where A is a tridiagonal matrix of order n, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

Error bounds on the solution and a condition estimate are also provided.

The routine ?gtsvx performs the following steps:

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 an upper triangular matrix with nonzeroes in only the main diagonal and first two superdiagonals.
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.
The system of equations is solved for X using the factored form of A.
Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

Input Parameters

fact	CHARACTER*1. Must be 'F' or 'N'. Specifies whether or not the factored form of the matrix `A` has been supplied on entry. If `fact = 'F'`: on entry, `dlf`, `df`, `duf`, `du2`, and `ipiv` contain the factored form of `A`; arrays `dl`, `d`, `du`, `dlf`, `df`, `duf`, `du2`, and `ipiv` will not be modified. If `fact = 'N'`, the matrix `A` will be copied to `dlf`, `df`, and `duf` and factored.
trans	CHARACTER1. Must be 'N', 'T', or 'C'. Specifies the form of the system of equations: If `trans = 'N'`, the system has the form `AX = B` (No transpose). If `trans = 'T'`, the system has the form `A`^T`X` = `B` (Transpose). If `trans = 'C'`, the system has the form `A`^H`X` = `B` (Conjugate transpose).
n	INTEGER. The number of linear equations, the order of the matrix `A`; `n`≥ 0.
nrhs	INTEGER. The number of right hand sides, the number of columns of the matrices `B` and `X`; `nrhs`≥ 0.
dl,d,du,dlf,df, duf,du2,b ,work	REAL for sgtsvx DOUBLE PRECISION for dgtsvx COMPLEX for cgtsvx DOUBLE COMPLEX for zgtsvx. Arrays: dl, size (`n` -1), contains the subdiagonal elements of `A`. d, size (`n`), contains the diagonal elements of `A`. du, size (`n` -1), contains the superdiagonal elements of `A`. dlf, size (`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 ?gttrf. df, size (`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`. duf, size (`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`. du2, size (`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`. b(ldb, ) contains the right-hand side matrix `B`. The second dimension of `b` must be at least `max(1, nrhs)`. work() is a workspace array. The size of work must be at least `max(1, 3n)` for real flavors and `max(1, 2n)` for complex flavors.
ldb	INTEGER. The leading dimension of `b`; `ldb≥ max(1, n)`.
ldx	INTEGER. The leading dimension of `x`; `ldx≥ max(1, n)`.
ipiv	INTEGER. Array, size at least `max(1, n)`. If `fact = 'F'`, then `ipiv` is an input argument and on entry contains the pivot indices, as returned by ?gttrf.
iwork	INTEGER. Workspace array, size (`n`). Used for real flavors only.
rwork	REAL for cgtsvx DOUBLE PRECISION for zgtsvx. Workspace array, size (`n`). Used for complex flavors only.

Output Parameters

x	REAL for sgtsvx DOUBLE PRECISION for dgtsvx COMPLEX for cgtsvx DOUBLE COMPLEX for zgtsvx. Array, size `ldx` by *. If `info = 0` or `info = n+1`, the array `x` contains the solution matrix `X`. The second dimension of `x` must be at least `max(1, nrhs)`.
dlf	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	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	If `fact = 'N'`, then `duf` is an output argument and on exit contains the `(n-1)` elements of the first superdiagonal of `U`.
du2	If `fact = 'N'`, then `du2` is an output argument and on exit contains the `(n-2)` elements of the second superdiagonal of `U`.
ipiv	The array `ipiv` is an output argument if `fact = 'N'`and, on exit, contains the pivot indices from the factorization `A = L*U` ; row `i` of the matrix was interchanged with row `ipiv`(`i`). The value of `ipiv`(`i`) will always be `i` or `i`+1; `ipiv`(`i`)=`i` indicates a row interchange was not required.
rcond	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. An 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains 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	REAL for single precision flavors DOUBLE PRECISION for double precision flavors. Array, size at least `max(1, nrhs)`. Contains the component-wise relative backward error for each solution vector `x(j)`, that is, the smallest relative change in any element of `A` or `B` that makes `x(j)` an exact solution.
info	INTEGER. If `info = 0`, the execution is successful. If `info = -i`, the `i`-th parameter had an illegal value. If `info = i`, and `i≤n`, then `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. If `info = i`, and `i = n + 1`, then `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.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine gtsvx interface are as follows:

dl	Holds the vector of length (`n-1`).
d	Holds the vector of length `n`.
du	Holds the vector of length (`n-1`).
b	Holds the matrix `B` of size (`n,nrhs`).
x	Holds the matrix `X` of size (`n,nrhs`).
dlf	Holds the vector of length (`n-1`).
df	Holds the vector of length `n`.
duf	Holds the vector of length (`n-1`).
du2	Holds the vector of length (`n-2`).
ipiv	Holds the vector of length `n`.
ferr	Holds the vector of length (`nrhs`).
berr	Holds the vector of length (`nrhs`).
fact	Must be 'N' or 'F'. The default value is 'N'. If `fact = 'F'`, then the arguments `dlf`, `df`, `duf`, `du2`, and `ipiv` must be present; otherwise, an error is returned.
trans	Must be 'N', 'C', or 'T'. The default value is 'N'.

Parent topic: LAPACK Linear Equation Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?gtsvx

Syntax

Include Files

Description

Input Parameters

Output Parameters

LAPACK 95 Interface Notes

See Also