Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?gttrs
Solves a system of linear equations with a tridiagonal coefficient matrix using the LU factorization computed by ?gttrf .
Syntax
call sgttrs ( trans , n , nrhs , dl , d , du , du2 , ipiv , b , ldb , info )
call dgttrs ( trans , n , nrhs , dl , d , du , du2 , ipiv , b , ldb , info )
call cgttrs ( trans , n , nrhs , dl , d , du , du2 , ipiv , b , ldb , info )
call zgttrs ( trans , n , nrhs , dl , d , du , du2 , ipiv , b , ldb , info )
call gttrs ( dl , d , du , du2 , b , ipiv [ , trans ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
sgttrs dgttrs cgttrs zgttrs gttrs
The routine solves for \(X\) in the following systems of linear equations with multiple right hand sides:
\(AX\) = \(B\) if trans = ‘N’ ,
\(A^{T}X = B\) if trans = ‘T’ ,
\(A^{H}X = B\) if trans = ‘C’ (for complex matrices only).
Before calling this routine, you must call ?gttrf (Computes the LU factorization of a tridiagonal matrix.) to compute the LU factorization of A .
Input Parameters
trans
CHARACTER*1 . Must be ‘N’ or ‘T’ or ‘C’ .
Indicates the form of the equations:
If trans = 'N' , then A*X = B is solved for X .
If trans = 'T' , then A:code:`T`*X = B is solved for X .
If trans = 'C' , then A:code:`H`*X = B is solved for X .
n
INTEGER . The order of A ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides, that is, the number of columns in B; nrhs≥ 0 .
dl , d , du , du2 , b
REAL for sgttrs
DOUBLE PRECISION for dgttrs
COMPLEX for cgttrs
DOUBLE COMPLEX for zgttrs .
Arrays: dl(n -1) , d(n) , du(n -1) , du2(n -2) , b(ldb,nrhs) .
The array dl contains the (n - 1) multipliers that define the matrix L from the LU factorization of A .
The array d contains the n diagonal elements of the upper triangular matrix U from the LU factorization of A .
The array du contains the ( n - 1) elements of the first superdiagonal of U .
The array du2 contains the ( n - 2) elements of the second superdiagonal of U .
The array b contains the matrix B whose columns are the right-hand sides for the systems of equations.
b
Array of size max(1, ldb * nrhs ) for column major layout and max(1, n * ldb ) for row major layout. Contains the matrix B whose columns are the right-hand sides for the systems of equations.
ldb
INTEGER . The leading dimension of b; ldb≥ max(1, n) .
ipiv
INTEGER . Array, size ( n ). The ipiv array, as returned by gttrf .
Output Parameters
b
Overwritten by the solution matrix X .
info
INTEGER .
If info=0 , the execution is successful.
If info = -i , the i -th parameter had an illegal value.
Return Values
No return value, info is an Output Parameter.
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 gttrs 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).
du2
Holds the vector of length ( n -2).
b
Holds the matrix B of size ( n , nrhs ).
ipiv
Holds the vector of length n .
trans
Must be ‘N’ , ‘C’ , or ‘T’ . The default value is ‘N’ .
Application Notes
For each right-hand side \(b\) , the computed solution is the exact solution of a perturbed system of equations \((A + E)x = b\) , where
\[|E| \le c(n) \varepsilon P|L||U|\]
\(c(n)\) is a modest linear function of \(k\) , and \(\varepsilon\) is the machine precision.
If \(x_{0}\) is the true solution, the computed solution \(x\) satisfies this error bound:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(kl + ku + 1) \operatorname{cond}(A, x) \varepsilon\]
where
\[\operatorname{cond(A,x)} = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
Note that \(cond(A,x)\) can be much smaller than \(\kappa_\infty (A)\) ; the condition number of \(A^{T}\) and \(A^{H}\) might or might not be equal to \(\kappa_\infty (A)\) .
The approximate number of floating-point operations for one right-hand side vector \(b\) is \(7n\) (including \(n\) divisions) for real flavors and \(34n\) (including \(2n\) divisions) for complex flavors.
To estimate the condition number \(\kappa_\infty (A)\) , call ?gtcon (Estimates the reciprocal of the condition number of a tridiagonal matrix.) .
To refine the solution and estimate the error, call ?gtrfs (Refines the solution of a system of linear equations with a tridiagonal coefficient matrix and estimates its error.) .