> For the complete documentation index, see [llms.txt](https://nelson-9.gitbook.io/nelson/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://nelson-9.gitbook.io/nelson/markdown/en/slicot/slicot_sb03od.md).

# slicot\_sb03od

Solution of stable continuous- or discrete-time Lyapunov equations (Cholesky factor).

## 📝 Syntax

* \[Q\_OUT, B\_OUT, SCALE, WR, WI, INFO] = slicot\_sb03od(DICO, FACT, TRANS, A, Q\_IN, B\_IN)

## 📥 Input argument

* DICO - Specifies the type of Lyapunov equation to be solved as follows: = 'C': continuous-time case; 'D': discrete-time case.
* FACT - Specifies whether or not the real Schur factorization of the matrix A is supplied on entry. = 'F': On entry, A and Q contain the factors from the real Schur factorization of the matrix A; = 'N': The Schur factorization of A will be computed and the factors will be stored in A and Q.
* TRANS - Specifies the form of op(K) to be used. = 'N': op(K) = K (No transpose); = 'T': op(K) = K\*\*T (Transpose).
* A - the leading N-by-N part of this array must contain the matrix A. If FACT = 'F', then A contains an upper quasi-triangular matrix S in Schur canonical form; the elements below the upper Hessenberg part of the array A are not referenced.
* Q\_IN - if FACT = 'F', then the leading N-by-N part of this array must contain the orthogonal matrix Q of the Schur factorization of A. Otherwise, Q need not be set on entry.
* B\_IN - if TRANS = 'N', and dimension (LDB,max(M,N)), if TRANS = 'T'. On entry, if TRANS = 'N', the leading M-by-N part of this array must contain the coefficient matrix B of the equation. On entry, if TRANS = 'T', the leading N-by-M part of this array must contain the coefficient matrix B of the equation.

## 📤 Output argument

* Q\_OUT - the leading N-by-N part of this array contains the orthogonal matrix Q of the Schur factorization of A. The contents of array Q is not modified if FACT = 'F'.
* B\_OUT - the leading N-by-N part of this array contains the upper triangular Cholesky factor U of the solution matrix X of the problem, X = op(U)'\*op(U). If M = 0 and N > 0, then U is set to zero.
* SCALE - The scale factor, scale, set less than or equal to 1 to prevent the solution overflowing.
* WR - If FACT = 'N', and INFO >= 0 and INFO less than 2, WR contains the real parts of the eigenvalues of A.
* WI - If FACT = 'N', and INFO >= 0 and INFO less than 2, WI contains the imaginary parts of the eigenvalues of A.
* INFO - = 0: successful exit.

## 📄 Description

To solve for X = op(U)'\*op(U) either the stable non-negative definite continuous-time Lyapunov equation or the convergent non-negative definite discrete-time Lyapunov equation.

## Used function(s)

SB03OD

## 📚 Bibliography

<http://slicot.org/objects/software/shared/doc/SB03OD.html>

## 💡 Example

```matlab
A = [-0.5000    0.5000         0;
         0         0         0;
   -0.5000         0    0.5000];
B_IN = [0.5000    1.5000    1.0000;
    1.0000    1.0000    1.0000;
    0.5000    1.0000    1.5000];
DICO = 'D';
FACT = 'N';
Q_IN = zeros(3, 3);
[Q_OUT, B_OUT, SCALE, WR, WI, INFO] = slicot_sb03od(DICO, FACT, TRANS, A, Q_IN, B_IN)
```

## 🕔 History

| Version | 📄 Description  |
| ------- | --------------- |
| 1.0.0   | initial version |