org.apache.commons.math3.linear

## Interface DecompositionSolver

-     public interface DecompositionSolver

  Interface handling decomposition algorithms that can solve A × X = B.

  Decomposition algorithms decompose an A matrix has a product of several specific matrices from which they can solve A × X = B in least squares sense: they find X such that ||A × X - B|| is minimal.

  Some solvers like `LUDecomposition` can only find the solution for square matrices and when the solution is an exact linear solution, i.e. when ||A × 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.

  Since:
    2.0

- - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `RealMatrix` | `getInverse()` Get the pseudo-inverse of the decomposed matrix.
    `boolean` | `isNonSingular()` Check if the decomposed matrix is non-singular.
    `RealMatrix` | `solve(RealMatrix b)` Solve the linear equation A × X = B for matrices A.
    `RealVector` | `solve(RealVector b)` Solve the linear equation A × X = B for matrices A.

- - ### Method Detail

    - #### solve

          RealVector solve(RealVector b)
                           throws SingularMatrixException

      Solve the linear equation A × X = B for matrices A.

      The A matrix is implicit, it is provided by the underlying decomposition algorithm.

      Parameters:
        `b` - right-hand side of the equation A × X = B
      Returns:
        a vector X that minimizes the two norm of A × X - B
      Throws:
        `DimensionMismatchException` - if the matrices dimensions do not match.
        `SingularMatrixException` - if the decomposed matrix is singular.

    - #### solve

          RealMatrix solve(RealMatrix b)
                           throws SingularMatrixException

      Solve the linear equation A × X = B for matrices A.

      The A matrix is implicit, it is provided by the underlying decomposition algorithm.

      Parameters:
        `b` - right-hand side of the equation A × X = B
      Returns:
        a matrix X that minimizes the two norm of A × X - B
      Throws:
        `DimensionMismatchException` - if the matrices dimensions do not match.
        `SingularMatrixException` - if the decomposed matrix is singular.

    - #### isNonSingular

          boolean isNonSingular()

      Check if the decomposed matrix is non-singular.
      Returns:
        true if the decomposed matrix is non-singular.

    - #### getInverse

          RealMatrix getInverse()
                                throws SingularMatrixException

      Get the pseudo-inverse of the decomposed matrix.

      This is equal to the inverse of the decomposed matrix, if such an inverse exists.

      If no such inverse exists, then the result has properties that resemble that of an inverse.

      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.

      Note however that some decompositions cannot compute a pseudo-inverse for all matrices. For example, the `LUDecomposition` is not defined for non-square matrices to begin with. The `QRDecomposition` can operate on non-square matrices, but will throw `SingularMatrixException` if the decomposed matrix is singular. Refer to the javadoc of specific decomposition implementations for more details.

      Returns:
        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

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