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 |
|---|---|
| public RealMatrix | Returns: pseudo-inverse matrix (which is the inverse, if it exists), if the decomposition can pseudo-invert the decomposed matrixGet the pseudo-inverse of the decomposed matrix. |
| public boolean | Returns: true if the decomposed matrix is non-singular.Check if the decomposed matrix is non-singular. |
| public RealVector | Returns: a vector X that minimizes the two norm of A × X - Bright-hand side of the equation A × X = B b)Solve the linear equation A × X = B for matrices A. |
| public RealMatrix | Returns: a matrix X that minimizes the two norm of A × X - Bright-hand side of the equation A × X = B b)Solve the linear equation A × X = B for matrices A. |
| getInverse | back to summary |
|---|---|
| public 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
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| isNonSingular | back to summary |
|---|---|
| public boolean isNonSingular() Check if the decomposed matrix is non-singular.
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| solve | back to summary |
|---|---|
| public 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.
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| solve | back to summary |
|---|---|
| public RealMatrix solve(final 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.
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