public interface DecompositionSolver
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.
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.
|
RealVector solve(RealVector b) throws SingularMatrixException
The A matrix is implicit, it is provided by the underlying decomposition algorithm.
b
- right-hand side of the equation A × X = BDimensionMismatchException
- if the matrices dimensions do not match.SingularMatrixException
- if the decomposed matrix is singular.RealMatrix solve(RealMatrix b) throws SingularMatrixException
The A matrix is implicit, it is provided by the underlying decomposition algorithm.
b
- right-hand side of the equation A × X = BDimensionMismatchException
- if the matrices dimensions do not match.SingularMatrixException
- if the decomposed matrix is singular.boolean isNonSingular()
RealMatrix getInverse() throws SingularMatrixException
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.
SingularMatrixException
- if the decomposed matrix is singular and the decomposition
can not compute a pseudo-inverseCopyright © 2003–2016 The Apache Software Foundation. All rights reserved.