cern.colt.matrix.linalg

## Class LUDecompositionQuick

• All Implemented Interfaces:
Serializable

```public class LUDecompositionQuick
extends Object
implements Serializable```
A low level version of `LUDecomposition`, avoiding unnecessary memory allocation and copying. The input to decompose methods is overriden with the result (LU). The input to solve methods is overriden with the result (X). In addition to LUDecomposition, this class also includes a faster variant of the decomposition, specialized for tridiagonal (and hence also diagonal) matrices, as well as a solver tuned for vectors. Its disadvantage is that it is a bit more difficult to use than LUDecomposition. Thus, you may want to disregard this class and come back later, if a need for speed arises.

An instance of this class remembers the result of its last decomposition. Usage pattern is as follows: Create an instance of this class, call a decompose method, then retrieve the decompositions, determinant, and/or solve as many equation problems as needed. Once another matrix needs to be LU-decomposed, you need not create a new instance of this class. Start again by calling a decompose method, then retrieve the decomposition and/or solve your equations, and so on. In case a LU matrix is already available, call method setLU instead of decompose and proceed with solving et al.

If a matrix shall not be overriden, use matrix.copy() and hand the the copy to methods.

For an m x n matrix A with m >= n, the LU decomposition is an m x n unit lower triangular matrix L, an n x n upper triangular matrix U, and a permutation vector piv of length m so that A(piv,:) = L*U; If m < n, then L is m x m and U is m x n.

The LU decomposition with pivoting always exists, even if the matrix is singular, so the decompose methods will never fail. The primary use of the LU decomposition is in the solution of square systems of simultaneous linear equations. Attempting to solve such a system will throw an exception if isNonsingular() returns false.

Serialized Form
• ### Field Summary

Fields
Modifier and Type Field and Description
`protected Algebra` `algebra`
`protected boolean` `isNonSingular`
`protected DoubleMatrix2D` `LU`
Array for internal storage of decomposition.
`protected int[]` `piv`
Internal storage of pivot vector.
`protected int` `pivsign`
pivot sign.
`protected int[]` `work1`
`protected int[]` `work2`
`protected double[]` `workDouble`
• ### Constructor Summary

Constructors
Constructor and Description
`LUDecompositionQuick()`
Constructs and returns a new LU Decomposition object with default tolerance 1.0E-9 for singularity detection.
`LUDecompositionQuick(double tolerance)`
Constructs and returns a new LU Decomposition object which uses the given tolerance for singularity detection;
• ### Method Summary

Methods
Modifier and Type Method and Description
`void` `decompose(DoubleMatrix2D A)`
Decomposes matrix A into L and U (in-place).
`void` ```decompose(DoubleMatrix2D A, int semiBandwidth)```
Decomposes the banded and square matrix A into L and U (in-place).
`double` `det()`
Returns the determinant, det(A).
`protected double[]` `getDoublePivot()`
Returns pivot permutation vector as a one-dimensional double array
`DoubleMatrix2D` `getL()`
Returns the lower triangular factor, L.
`DoubleMatrix2D` `getLU()`
Returns a copy of the combined lower and upper triangular factor, LU.
`int[]` `getPivot()`
Returns the pivot permutation vector (not a copy of it).
`DoubleMatrix2D` `getU()`
Returns the upper triangular factor, U.
`boolean` `isNonsingular()`
Returns whether the matrix is nonsingular (has an inverse).
`protected boolean` `isNonsingular(DoubleMatrix2D matrix)`
Returns whether the matrix is nonsingular.
`protected DoubleMatrix2D` `lowerTriangular(DoubleMatrix2D A)`
Modifies the matrix to be a lower triangular matrix.
`protected int` `m()`
`protected int` `n()`
`void` `setLU(DoubleMatrix2D LU)`
Sets the combined lower and upper triangular factor, LU.
`void` `solve(DoubleMatrix1D B)`
Solves the system of equations A*X = B (in-place).
`void` `solve(DoubleMatrix2D B)`
Solves the system of equations A*X = B (in-place).
`String` `toString()`
Returns a String with (propertyName, propertyValue) pairs.
`protected DoubleMatrix2D` `upperTriangular(DoubleMatrix2D A)`
Modifies the matrix to be an upper triangular matrix.
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait`
• ### Field Detail

• #### LU

`protected DoubleMatrix2D LU`
Array for internal storage of decomposition.
• #### pivsign

`protected int pivsign`
pivot sign.
• #### piv

`protected int[] piv`
Internal storage of pivot vector.
• #### isNonSingular

`protected boolean isNonSingular`
• #### algebra

`protected Algebra algebra`
• #### workDouble

`protected transient double[] workDouble`
• #### work1

`protected transient int[] work1`
• #### work2

`protected transient int[] work2`
• ### Constructor Detail

• #### LUDecompositionQuick

`public LUDecompositionQuick()`
Constructs and returns a new LU Decomposition object with default tolerance 1.0E-9 for singularity detection.
• #### LUDecompositionQuick

`public LUDecompositionQuick(double tolerance)`
Constructs and returns a new LU Decomposition object which uses the given tolerance for singularity detection;
• ### Method Detail

• #### decompose

`public void decompose(DoubleMatrix2D A)`
Decomposes matrix A into L and U (in-place). Upon return A is overridden with the result LU, such that L*U = A. Uses a "left-looking", dot-product, Crout/Doolittle algorithm.
Parameters:
`A` - any matrix.
• #### decompose

```public void decompose(DoubleMatrix2D A,
int semiBandwidth)```
Decomposes the banded and square matrix A into L and U (in-place). Upon return A is overridden with the result LU, such that L*U = A. Currently supports diagonal and tridiagonal matrices, all other cases fall through to `decompose(DoubleMatrix2D)`.
Parameters:
`semiBandwidth` - == 1 --> A is diagonal, == 2 --> A is tridiagonal.
`A` - any matrix.
• #### det

`public double det()`
Returns the determinant, det(A).
Throws:
`IllegalArgumentException` - if A.rows() != A.columns() (Matrix must be square).
• #### getDoublePivot

`protected double[] getDoublePivot()`
Returns pivot permutation vector as a one-dimensional double array
Returns:
(double) piv
• #### getL

`public DoubleMatrix2D getL()`
Returns the lower triangular factor, L.
Returns:
L
• #### getLU

`public DoubleMatrix2D getLU()`
Returns a copy of the combined lower and upper triangular factor, LU.
Returns:
LU
• #### getPivot

`public int[] getPivot()`
Returns the pivot permutation vector (not a copy of it).
Returns:
piv
• #### getU

`public DoubleMatrix2D getU()`
Returns the upper triangular factor, U.
Returns:
U
• #### isNonsingular

`public boolean isNonsingular()`
Returns whether the matrix is nonsingular (has an inverse).
Returns:
true if U, and hence A, is nonsingular; false otherwise.
• #### isNonsingular

`protected boolean isNonsingular(DoubleMatrix2D matrix)`
Returns whether the matrix is nonsingular.
Returns:
true if matrix is nonsingular; false otherwise.
• #### lowerTriangular

`protected DoubleMatrix2D lowerTriangular(DoubleMatrix2D A)`
Modifies the matrix to be a lower triangular matrix.

Examples:

 3 x 5 matrix: 9, 9, 9, 9, 9 9, 9, 9, 9, 9 9, 9, 9, 9, 9 triang.Upper ==> 3 x 5 matrix: 9, 9, 9, 9, 9 0, 9, 9, 9, 9 0, 0, 9, 9, 9 5 x 3 matrix: 9, 9, 9 9, 9, 9 9, 9, 9 9, 9, 9 9, 9, 9 triang.Upper ==> 5 x 3 matrix: 9, 9, 9 0, 9, 9 0, 0, 9 0, 0, 0 0, 0, 0 3 x 5 matrix: 9, 9, 9, 9, 9 9, 9, 9, 9, 9 9, 9, 9, 9, 9 triang.Lower ==> 3 x 5 matrix: 1, 0, 0, 0, 0 9, 1, 0, 0, 0 9, 9, 1, 0, 0 5 x 3 matrix: 9, 9, 9 9, 9, 9 9, 9, 9 9, 9, 9 9, 9, 9 triang.Lower ==> 5 x 3 matrix: 1, 0, 0 9, 1, 0 9, 9, 1 9, 9, 9 9, 9, 9
Returns:
A (for convenience only).
`#triangulateUpper(DoubleMatrix2D)`
• #### m

`protected int m()`
• #### n

`protected int n()`
• #### setLU

`public void setLU(DoubleMatrix2D LU)`
Sets the combined lower and upper triangular factor, LU. The parameter is not checked; make sure it is indeed a proper LU decomposition.
• #### solve

`public void solve(DoubleMatrix1D B)`
Solves the system of equations A*X = B (in-place). Upon return B is overridden with the result X, such that L*U*X = B(piv).
Parameters:
`B` - A vector with B.size() == A.rows().
Throws:
`IllegalArgumentException` - if B.size() != A.rows().
`IllegalArgumentException` - if A is singular, that is, if !isNonsingular().
`IllegalArgumentException` - if A.rows() < A.columns().
• #### solve

`public void solve(DoubleMatrix2D B)`
Solves the system of equations A*X = B (in-place). Upon return B is overridden with the result X, such that L*U*X = B(piv,:).
Parameters:
`B` - A matrix with as many rows as A and any number of columns.
Throws:
`IllegalArgumentException` - if B.rows() != A.rows().
`IllegalArgumentException` - if A is singular, that is, if !isNonsingular().
`IllegalArgumentException` - if A.rows() < A.columns().
• #### toString

`public String toString()`
Returns a String with (propertyName, propertyValue) pairs. Useful for debugging or to quickly get the rough picture. For example,
```rank          : 3
trace         : 0
```
Overrides:
`toString` in class `Object`
• #### upperTriangular

`protected DoubleMatrix2D upperTriangular(DoubleMatrix2D A)`
Modifies the matrix to be an upper triangular matrix.
Returns:
A (for convenience only).
`#triangulateLower(DoubleMatrix2D)`