public class LU extends Object implements Serializable, Cloneable
double
.
LU
performs an LU factorization of a
real general coefficient matrix. The condition
method
estimates the reciprocal of the condition number of
the matrix. The LU factorization is done using scaled partial pivoting.
Scaled partial pivoting differs from partial pivoting in that the pivoting
strategy is the same as if each row were scaled to have the same infinity
norm.
The condition number of the matrix A is defined to be . Since it is expensive to compute , the condition number is only estimated. The estimation algorithm is the same as used by LINPACK and is described in a paper by Cline et al. (1979).
Note that A is not retained for use by other methods of this
class, only the factorization of A is retained. Thus, A
is a required parameter to the condition
method.
An estimated condition number greater than
(where is machine precision) indicates that
very small changes in A can cause very large changes
in the solution x. Iterative refinement can sometimes
find the solution to such a system. If there is conern about the input
matrix being illconditioned, the user of this class should check the
condition number of the input matrix using the condition
method
before using one of the other class methods.
LU
fails if U, the upper triangular
part of the factorization, has a zero diagonal element. This can occur only
if A either is singular or is very close to a singular
matrix.
Use the solve
method to solve systems of equations. The
determinant
method can be called to compute the determinant of
the coefficient matrix.
LU
is based on the LINPACK routine SGECO
; see
Dongarra et al. (1979). SGECO
uses unscaled partial pivoting.
Modifier and Type  Field and Description 

protected double[][] 
factor
This is an n by n matrix containing the LU factorization
of the matrix A.

protected int[] 
ipvt
Vector of length n containing the pivot sequence for the factorization.

Constructor and Description 

LU(double[][] a)
Creates the LU factorization of a square matrix of type
double . 
Modifier and Type  Method and Description 

double 
condition(double[][] a)
Return an estimate of the reciprocal of the condition number
of a matrix.

double 
determinant()
Return the determinant of the matrix used to construct this instance.

double[][] 
getL()
Returns the lower triangular portion of the LU
factorization of A.

double[][] 
getPermutationMatrix()
Returns the permutation matrix which results from the
LU factorization of A.

double[][] 
getU()
Returns the unit upper triangular portion of the LU
factorization of A.

double[][] 
inverse()
Returns the inverse of the matrix used to construct this instance.

double[] 
solve(double[] b)
Return the solution x of the linear system Ax = b using the
LU factorization of A.

static double[] 
solve(double[][] a,
double[] b)
Solve Ax = b for x using the LU factorization of A.

double[] 
solveTranspose(double[] b)
Return the solution x of the linear system .

protected double[][] factor
protected int[] ipvt
public LU(double[][] a) throws SingularMatrixException
double
.a
 the double
square matrix to be factoredIllegalArgumentException
 is thrown when
the row lengths of input matrix are not equal
(for example, the matrix edges are "jagged".)SingularMatrixException
 is thrown when
the input matrix is singular.public double condition(double[][] a)
a
 the double
square matrix for which
the reciprocal of the condition number is desireddouble
value representing an estimate of the
reciprocal of the condition number of the matrixpublic double determinant()
double
scalar containing the determinant of the
matrix used to construct this instancepublic double[][] getL()
Scaled partial pivoting is used to achieve the LU
factorization. The resulting factorization is such that
, where A is the input matrix
a
, P is the permutation matrix
returned by getPermutationMatrix
, L is the lower
triangular matrix returned by getL
, and U is the unit
upper triangular matrix returned by getU
.
double
matrix containing L, the lower triangular
portion of the LU factorization of A.public double[][] getPermutationMatrix()
Scaled partial pivoting is used to achieve the LU
factorization. The resulting factorization is such that
, where A is the input matrix
a
, P is the permutation matrix returned by
getPermutationMatrix
, L is the lower triangular
matrix returned by getL
, and U is the unit upper
triangular matrix returned by getU
.
double
matrix containing the permuted
identity matrix as a result of the LU factorization
of A.public double[][] getU()
Scaled partial pivoting is used to achieve the LU
factorization. The resulting factorization is such that
, where A is the input matrix
a
, P is the permutation matrix returned by
getPermutationMatrix
, L is the lower triangular
matrix returned by getL
, and U is the unit upper
triangular matrix returned by getU
.
double
matrix containing U, the unit upper
triangular portion of the LU factorization of A.public double[][] inverse()
double
matrix representing the inverse
of the matrix used to construct this instancepublic double[] solve(double[] b)
b
 a double
array containing the righthand side of the
linear systemdouble
array containing the solution to the linear
system of equationspublic static double[] solve(double[][] a, double[] b) throws SingularMatrixException
a
 a double
square matrixb
 a double
column vectordouble
column vector containing the
solution to the linear system of equationsIllegalArgumentException
 This exception is thrown when (1) the lengths of the
rows of the input matrix are not uniform, and (2) the
number of rows in the input matrix is not equal to the
number of elements in x.SingularMatrixException
 is thrown when the matrix is singular.public double[] solveTranspose(double[] b)
b
 double
array containing the righthand side of the linear systemdouble
array containing the solution to the linear system of
equationsCopyright © 19702015 Rogue Wave Software
Built October 13 2015.