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 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
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package org.apache.commons.math3.linear;

/**
 * Interface handling decomposition algorithms that can solve A &times; X = B.
 * <p>
 * Decomposition algorithms decompose an A matrix has a product of several specific
 * matrices from which they can solve A &times; X = B in least squares sense: they find X
 * such that ||A &times; X - B|| is minimal.
 * <p>
 * Some solvers like {@link LUDecomposition} can only find the solution for
 * square matrices and when the solution is an exact linear solution, i.e. when
 * ||A &times; X - B|| is exactly 0. Other solvers can also find solutions
 * with non-square matrix A and with non-null minimal norm. If an exact linear
 * solution exists it is also the minimal norm solution.
 *
 * @version $Id: DecompositionSolver.java 1566017 2014-02-08 14:13:34Z tn $
 * @since 2.0
 */
public interface DecompositionSolver {

    /**
     * Solve the linear equation A &times; X = B for matrices A.
     * <p>
     * The A matrix is implicit, it is provided by the underlying
     * decomposition algorithm.
     *
     * @param b right-hand side of the equation A &times; X = B
     * @return a vector X that minimizes the two norm of A &times; X - B
     * @throws org.apache.commons.math3.exception.DimensionMismatchException
     * if the matrices dimensions do not match.
     * @throws SingularMatrixException if the decomposed matrix is singular.
     */
    RealVector solve(final RealVector b) throws SingularMatrixException;

    /**
     * Solve the linear equation A &times; X = B for matrices A.
     * <p>
     * The A matrix is implicit, it is provided by the underlying
     * decomposition algorithm.
     *
     * @param b right-hand side of the equation A &times; X = B
     * @return a matrix X that minimizes the two norm of A &times; X - B
     * @throws org.apache.commons.math3.exception.DimensionMismatchException
     * if the matrices dimensions do not match.
     * @throws SingularMatrixException if the decomposed matrix is singular.
     */
    RealMatrix solve(final RealMatrix b) throws SingularMatrixException;

    /**
     * Check if the decomposed matrix is non-singular.
     * @return true if the decomposed matrix is non-singular.
     */
    boolean isNonSingular();

    /**
     * Get the <a href="http://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_pseudoinverse">pseudo-inverse</a>
     * of the decomposed matrix.
     * <p>
     * <em>This is equal to the inverse  of the decomposed matrix, if such an inverse exists.</em>
     * <p>
     * If no such inverse exists, then the result has properties that resemble that of an inverse.
     * <p>
     * In particular, in this case, if the decomposed matrix is A, then the system of equations
     * \( A x = b \) may have no solutions, or many. If it has no solutions, then the pseudo-inverse
     * \( A^+ \) gives the "closest" solution \( z = A^+ b \), meaning \( \left \| A z - b \right \|_2 \)
     * is minimized. If there are many solutions, then \( z = A^+ b \) is the smallest solution,
     * meaning \( \left \| z \right \|_2 \) is minimized.
     * <p>
     * Note however that some decompositions cannot compute a pseudo-inverse for all matrices.
     * For example, the {@link LUDecomposition} is not defined for non-square matrices to begin
     * with. The {@link QRDecomposition} can operate on non-square matrices, but will throw
     * {@link SingularMatrixException} if the decomposed matrix is singular. Refer to the javadoc
     * of specific decomposition implementations for more details.
     *
     * @return pseudo-inverse matrix (which is the inverse, if it exists),
     * if the decomposition can pseudo-invert the decomposed matrix
     * @throws SingularMatrixException if the decomposed matrix is singular and the decomposition
     * can not compute a pseudo-inverse
     */
    RealMatrix getInverse() throws SingularMatrixException;
}
