Contents

# Systems of Linear Equations: ScaLAPACK Computational Routines

ScaLAPACK supports routines for the systems of equations with the following types of matrices:
• general
• general banded
• general diagonally dominant-like banded (including general tridiagonal)
• symmetric or Hermitian positive-definite
• symmetric or Hermitian positive-definite banded
• symmetric or Hermitian positive-definite tridiagonal
A diagonally dominant-like matrix is defined as a matrix for which it is known in advance that pivoting is not required in the
LU
factorization of this matrix.
For the above matrix types, the library includes routines for performing the following computations:
factoring
the matrix;
equilibrating
the matrix;
solving
a system of linear equations;
estimating the condition number
of a matrix;
refining
the solution of linear equations and computing its error bounds;
inverting
the matrix. Note that for some of the listed matrix types only part of the computational routines are provided (for example, routines that refine the solution are not provided for band or tridiagonal matrices). See Table
“Computational Routines for Systems of Linear Equations”
for full list of available routines.
To solve a particular problem, you can either call two or more computational routines or call a corresponding driver routine that combines several tasks in one call. Thus, to solve a system of linear equations with a general matrix, you can first call
p?getrf
(
LU
factorization) and then
p?getrs
(computing the solution). Then, you might wish to call
p?gerfs
to refine the solution and get the error bounds. Alternatively, you can just use the driver routine
p?gesvx
which performs all these tasks in one call.
Table
“Computational Routines for Systems of Linear Equations”
lists the ScaLAPACK computational routines for factorizing, equilibrating, and inverting matrices, estimating their condition numbers, solving systems of equations with real matrices, refining the solution, and estimating its error.
Computational Routines for Systems of Linear Equations
Matrix type, storage scheme
Factorize matrix
Equilibrate matrix
Solve system
Condition number
Estimate error
Invert matrix
general (partial pivoting)
general band (partial pivoting)

general band (no pivoting)

general tridiagonal (no pivoting)

symmetric/Hermitian positive-definite
symmetric/Hermitian positive-definite, band

symmetric/Hermitian positive-definite, tridiagonal

triangular

In this table
?
stands for
s
(single precision real),
d
(double precision real),
c
(single precision complex), or
z
(double precision complex).

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