Balances a general matrix to improve the accuracy of computed eigenvalues and eigenvectors.
Use the Back Transform Eigenvectors node after balancing a matrix with the Matrix Balance node and computing the eigenvectors of the balanced matrix with the Eigenvalues and Vectors node. Use the outputs of the Matrix Balance node and the Eigenvalues and Vectors node as the inputs of the Back Transform Eigenvectors node.
A real general matrix.
This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.
Type of matrix balance operation to perform.
Name | Value | Description |
---|---|---|
Neither Permuted nor Scaled | 0 | The node neither permutes nor scales the matrix. |
Permuted but not Scaled | 1 | The node permutes but does not scale the matrix. |
Scaled but not Permuted | 2 | The node scales but does not permute the matrix. |
Both Permuted and Scaled | 3 | The node permutes and scales the matrix. |
Default: Both Permuted and Scaled
Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.
Default: No error
The balanced matrix containing the same eigenvalues as the input matrix.
The form of the balanced matrix.
If job is Neither Permuted nor Scaled or Scaled but not Permuted, this output equals 0.
Algorithm for Defining the Form of the Balanced Matrix
balanced matrix_{(i,j)} = 0 if i > j and 0 ≤ j < index low.
The form of the balanced matrix.
If job is Neither Permuted nor Scaled or Scaled but not Permuted, this output equals n - 1.
Algorithm for Defining the Form of the Balanced Matrix
balanced matrix_{(i,j)} = 0 if i > j and index high < i ≤ n - 1.
Details about the permutations and scaling factors.
Algorithm for Calculating scale
If p_{j} is the index of the row and column interchanged with row and column j, and d_{j} is the scaling factor used to balance row and column j, the following equations define how this node computes the values of this output.
where i_{low} is the output index low and i_{high} is the output index high.
You can use one or both of the following similarity transformations to balance a matrix A and improve the accuracy of computed eigenvalues and eigenvectors:
The following expression defines the permutation of matrix A to block upper triangular form.
where
The diagonal elements of A'_{11} and A'_{33} are eigenvalues of A. The central diagonal block A'_{22} starts from column(row) input index low and ends in column(row) input index high of A'. If no suitable permutation of A exists, the following conditions are true:
The following expression defines the scaling of matrix A' to reduce the norm of matrix A'_{22}.
so that ||A"_{22}|| < ||A'_{22}||, which reduces the effect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.
Where This Node Can Run:
Desktop OS: Windows
FPGA: