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package org.apache.commons.math3.linear;
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
/**
* Interface handling decomposition algorithms that can solve A × X = B.
* <p>
* 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.
* <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 × 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
*/
public interface DecompositionSolver {
Solve the linear equation A × X = B for matrices A.
The A matrix is implicit, it is provided by the underlying
decomposition algorithm.
Params: - b – right-hand side of the equation A × X = B
Throws: - DimensionMismatchException –
if the matrices dimensions do not match.
- SingularMatrixException – if the decomposed matrix is singular.
Returns: a vector X that minimizes the two norm of A × X - B
/**
* Solve the linear equation A × 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 × X = B
* @return a vector X that minimizes the two norm of A × 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 × X = B for matrices A.
The A matrix is implicit, it is provided by the underlying
decomposition algorithm.
Params: - b – right-hand side of the equation A × X = B
Throws: - DimensionMismatchException –
if the matrices dimensions do not match.
- SingularMatrixException – if the decomposed matrix is singular.
Returns: a matrix X that minimizes the two norm of A × X - B
/**
* Solve the linear equation A × 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 × X = B
* @return a matrix X that minimizes the two norm of A × 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.
Returns: true if the decomposed matrix is non-singular.
/**
* Check if the decomposed matrix is non-singular.
* @return true if the decomposed matrix is non-singular.
*/
boolean isNonSingular();
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.
Throws: - SingularMatrixException – if the decomposed matrix is singular and the decomposition
can not compute a pseudo-inverse
Returns: pseudo-inverse matrix (which is the inverse, if it exists),
if the decomposition can pseudo-invert the decomposed matrix
/**
* 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;
}