1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
10   *
11   * Unless required by applicable law or agreed to in writing, software
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17
18  package org.apache.commons.math3.linear;
19
20  /**
21   * Interface handling decomposition algorithms that can solve A &times; X = B.
22   * <p>
23   * Decomposition algorithms decompose an A matrix has a product of several specific
24   * matrices from which they can solve A &times; X = B in least squares sense: they find X
25   * such that ||A &times; X - B|| is minimal.
26   * <p>
27   * Some solvers like {@link LUDecomposition} can only find the solution for
28   * square matrices and when the solution is an exact linear solution, i.e. when
29   * ||A &times; X - B|| is exactly 0. Other solvers can also find solutions
30   * with non-square matrix A and with non-null minimal norm. If an exact linear
31   * solution exists it is also the minimal norm solution.
32   *
33   * @since 2.0
34   */
35  public interface DecompositionSolver {
36
37      /**
38       * Solve the linear equation A &times; X = B for matrices A.
39       * <p>
40       * The A matrix is implicit, it is provided by the underlying
41       * decomposition algorithm.
42       *
43       * @param b right-hand side of the equation A &times; X = B
44       * @return a vector X that minimizes the two norm of A &times; X - B
45       * @throws org.apache.commons.math3.exception.DimensionMismatchException
46       * if the matrices dimensions do not match.
47       * @throws SingularMatrixException if the decomposed matrix is singular.
48       */
49      RealVector solve(final RealVector b) throws SingularMatrixException;
50
51      /**
52       * Solve the linear equation A &times; X = B for matrices A.
53       * <p>
54       * The A matrix is implicit, it is provided by the underlying
55       * decomposition algorithm.
56       *
57       * @param b right-hand side of the equation A &times; X = B
58       * @return a matrix X that minimizes the two norm of A &times; X - B
59       * @throws org.apache.commons.math3.exception.DimensionMismatchException
60       * if the matrices dimensions do not match.
61       * @throws SingularMatrixException if the decomposed matrix is singular.
62       */
63      RealMatrix solve(final RealMatrix b) throws SingularMatrixException;
64
65      /**
66       * Check if the decomposed matrix is non-singular.
67       * @return true if the decomposed matrix is non-singular.
68       */
69      boolean isNonSingular();
70
71      /**
72       * Get the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a>
73       * of the decomposed matrix.
74       * <p>
75       * <em>This is equal to the inverse  of the decomposed matrix, if such an inverse exists.</em>
76       * <p>
77       * If no such inverse exists, then the result has properties that resemble that of an inverse.
78       * <p>
79       * In particular, in this case, if the decomposed matrix is A, then the system of equations
80       * $$A x = b$$ may have no solutions, or many. If it has no solutions, then the pseudo-inverse
81       * $$A^+$$ gives the "closest" solution $$z = A^+ b$$, meaning $$\left \| A z - b \right \|_2$$
82       * is minimized. If there are many solutions, then $$z = A^+ b$$ is the smallest solution,
83       * meaning $$\left \| z \right \|_2$$ is minimized.
84       * <p>
85       * Note however that some decompositions cannot compute a pseudo-inverse for all matrices.
86       * For example, the {@link LUDecomposition} is not defined for non-square matrices to begin
87       * with. The {@link QRDecomposition} can operate on non-square matrices, but will throw
88       * {@link SingularMatrixException} if the decomposed matrix is singular. Refer to the javadoc
89       * of specific decomposition implementations for more details.
90       *
91       * @return pseudo-inverse matrix (which is the inverse, if it exists),
92       * if the decomposition can pseudo-invert the decomposed matrix
93       * @throws SingularMatrixException if the decomposed matrix is singular and the decomposition
94       * can not compute a pseudo-inverse
95       */
96      RealMatrix getInverse() throws SingularMatrixException;
97  }