Operators

# decompose_matrix (Operator)

## Name

decompose_matrix — Decompose a matrix.

## Signature

decompose_matrix( : : MatrixID, MatrixType : Matrix1ID, Matrix2ID)

## Description

The operator decompose_matrix decomposes the square input Matrix given by the matrix handle MatrixID. The results are stored in two generated matrices Matrix1 and Matrix2. The operator returns the matrix handles Matrix1ID and Matrix2ID. Access to the elements of the matrices is possible e.g. with the operator get_full_matrix.

The type of the input Matrix can be selected via the parameter MatrixType. The following values are supported: 'general' for general, 'symmetric' for symmetric, 'positive_definite' for symmetric positive definite, and 'tridiagonal' for tridiagonal matrices.

The decomposition MatrixType = 'general' or 'tridiagonal' is a LU factorization (Lower/Upper) with the form

```  Matrix = Matrix1 * Matrix2
```

The output Matrix1 is a lower triangular matrix with unit diagonal elements and interchanged rows. The output Matrix2 is an upper triangular matrix.

Example for a factorization of a general matrix:

```
/   5.0   -3.0    1.0  \
Matrix = |   0.0    2.0   -1.0  |    ->
\  -5.0   -1.0    5.0  /

/    1     0    0  \             /  5.0  -3.0   1.0  \
Matrix1 = |   0.0  -0.5   1  |   Matrix2 = |   0   -4.0   6.0  |
\  -1.0    1    0  /             \   0     0    2.0  /
```

Example for a factorization of a tridiagonal matrix:

```
/  -8.0  -8.0   0.0   0.0  \
|  -8.0   6.0  -4.0   0.0  |
Matrix = |   0.0   7.0  -2.0   7.0  |    ->
\   0.0   0.0   5.0  -3.0  /

/   1    0    0   0  \               / -8.0   -8.0   0.0   0.0  \
|  1.0   1    0   0  |               |   0    14.0  -4.0   0.0  |
Matrix1 = |  0.0  0.5  0.0  1  |     Matrix2 = |   0      0    5.0  -3.0  |
\  0.0  0.0   1   0  /               \   0      0     0    7.0  /
```

For MatrixType = 'symmetric' the factorization is a UDU^T decomposition (Upper/Diagonal/Upper) with the form

```  Matrix = Matrix1 * Matrix2 *
Matrix1^T.
```

where the output Matrix1 is an upper triangular matrix with interchanaged columns. The output matrix Matrix2 is a symmetric block diagonal matrix with 1 x 1 and 2 x 2 diagonal blocks.

Example for a factorization of a symmetric matrix:

```
/   3.0  -2.0   7.0  -1.0  \
|  -2.0  -2.0   4.0   0.0  |
Matrix = |   7.0   4.0   8.0   1.0  |    ->
\  -1.0   0.0   1.0   0.0  /

/  0    0     1    0.0  \               / 27.0   0    0     0  \
|  0    1    0.0   2.0  |               |   0  -2.0   0     0  |
Matrix1 = |  1  -1.0  -1.0 -10.0  |     Matrix2 = |   0    0   3.0  -1.0 |
\  0    0     0     1   /               \   0    0  -1.0   0.0 /
```

For MatrixType = 'positive_definite' a Cholesky factorization is computed with the form

```  Matrix = Matrix1 * Matrix2
```

where the output Matrix1 is a lower triangular matrix and the output matrix Matrix2 is an upper triangular matrix. Furthermore, the Matrix2 is the transpose of the matrix Matrix1.

Example for a factorization of a positive definite matrix:

```
/   9.0   12.0   -6.0  \
Matrix = |  12.0   17.0   -7.0  |    ->
\  -6.0   -7.0   14.0  /

/   3.0   0    0   \                /  3.0  4.0  -2.0  \
Matrix1 = |   4.0  1.0   0   |      Matrix2 = |   0   1.0   1.0  |
\  -2.0  1.0  3.0  /                \   0    0    3.0  /
```

It should be noted that in the examples there are differences in the meaning of the values of the output matrices: If a value is shown as an integer number, e.g., 0 or 1, the value of this element is per definition this certain value. If the number is shown as a floating point number, e.g., 0.0 or 1.0, the value is computed by the operator.

## Attention

For MatrixType = 'symmetric' or 'positive_definite', the upper triangular part of the input Matrix must contain the relevant information of the matrix. The strictly lower triangular part of the matrix is not referenced. For MatrixType = 'tridiagonal', only the main diagonal, the superdiagonal, and the subdiagonal of the input Matrix are used. The other parts of the matrix are not referenced. If the referenced part of the input Matrix is not of the specified type, an exception is raised.

## Parallelization

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Processed without parallelization.

## Parameters

MatrixID (input_control)  matrix (integer)

Matrix handle of the input matrix.

MatrixType (input_control)  string (string)

Type of the input matrix.

Default value: 'general'

List of values: 'general', 'positive_definite', 'symmetric', 'tridiagonal'

Matrix1ID (output_control)  matrix (integer)

Matrix handle with the output matrix 1.

Matrix2ID (output_control)  matrix (integer)

Matrix handle with the output matrix 2.

## Result

If the parameters are valid, the operator decompose_matrix returns the value 2 (H_MSG_TRUE). If necessary, an exception is raised.

## References

David Poole: “Linear Algebra: A Modern Introduction”; Thomson; Belmont; 2006.
Gene H. Golub, Charles F. van Loan: “Matrix Computations”; The Johns Hopkins University Press; Baltimore and London; 1996.

## Module

Foundation

 Operators