Developer Reference for Intel® oneAPI Math Kernel Library for Fortran
?tptrs
Solves a system of linear equations with a packed triangular coefficient matrix, with multiple right-hand sides.
Syntax
call stptrs ( uplo , trans , diag , n , nrhs , ap , b , ldb , info )
call dtptrs ( uplo , trans , diag , n , nrhs , ap , b , ldb , info )
call ctptrs ( uplo , trans , diag , n , nrhs , ap , b , ldb , info )
call ztptrs ( uplo , trans , diag , n , nrhs , ap , b , ldb , info )
call tptrs ( ap , b [ , uplo ] [ , trans ] [ , diag ] [ , info ] )
Include Files
mkl.fi , mkl_lapack.f90
Description
stptrs dtptrs ctptrs ztptrs tptrs
The routine solves for X the following systems of linear equations with a packed triangular matrix A , with multiple right-hand sides stored in B :
A*X = B
if trans = ‘N’ ,
A^{T}*X = B
if trans = ‘T’ ,
A:code:`H`*X = B
if trans = ‘C’ (for complex matrices only).
Input Parameters
uplo
CHARACTER*1 . Must be ‘U’ or ‘L’ .
Indicates whether A is upper or lower triangular:
If uplo = 'U' , then A is upper triangular.
If uplo = 'L' , then A is lower triangular.
trans
CHARACTER*1 . Must be ‘N’ or ‘T’ or ‘C’ .
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 .
diag
CHARACTER*1 . Must be ‘N’ or ‘U’ .
If diag = 'N' , then A is not a unit triangular matrix.
If diag = 'U' , then A is unit triangular: diagonal elements are assumed to be 1 and not referenced in the array ap .
n
INTEGER . The order of A ; the number of rows in B ; n ≥ 0.
nrhs
INTEGER . The number of right-hand sides; nrhs ≥ 0.
ap
REAL for stptrs
DOUBLE PRECISION for dtptrs
COMPLEX for ctptrs
DOUBLE COMPLEX for ztptrs .
The dimension of array ap(*) must be at least max(1, n(n+1)/2). The array ap contains the matrix A in packed storage (see Matrix Storage Schemes ).
b
REAL for stptrs
DOUBLE PRECISION for dtptrs
COMPLEX for ctptrs
DOUBLE COMPLEX for ztptrs .
The array b(ldb,*) contains the matrix B whose columns are the right-hand sides for the system of equations.
The second dimension of b must be at least max(1, nrhs) .
ldb
INTEGER . The leading dimension of b ; ldb≥ max(1, n) .
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 tptrs interface are as follows:
ap
Holds the array A of size (n*(n+1)/2) .
b
Holds the matrix B of size ( n , nrhs ).
uplo
Must be ‘U’ or ‘L’ . The default value is ‘U’ .
trans
Must be ‘N’ , ‘C’ , or ‘T’ . The default value is ‘N’ .
diag
Must be ‘N’ or ‘U’ . 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
\[|E| \leq c(n) \varepsilon |A|\]
\(c(n)\) 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:
\[\frac{\|x - x_0\|_\infty}{\|x\|_\infty} \leq c(n) \operatorname{cond}(A, x) \varepsilon, \text{ provided } c(n) \operatorname{cond}(A,x)\varepsilon < 1\]
where
\[\operatorname{cond}(A,x) = || |A^{-1}| |A| |x|_\infty || / ||x||_\infty \leq ||A^{-1}||_\infty ||A||_\infty = \kappa_\infty(A).\]
The approximate number of floating-point operations for one right-hand side vector \(b\) is \(n^2\) for real flavors and \(4n^{2}\) for complex flavors.
To estimate the condition number \(\kappa_{\infty}(A)\) , call ?tpcon (Estimates the reciprocal of the condition number of a packed triangular matrix.) .
To estimate the error in the solution, call ?tprfs (Estimates the error in the solution of a system of linear equations with a packed triangular coefficient matrix.) .