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### 18.4 Functions of a Matrix

Function File: expm (A)

Return the exponential of a matrix.

The matrix exponential is defined as the infinite Taylor series

```expm (A) = I + A + A^2/2! + A^3/3! + …
```

However, the Taylor series is not the way to compute the matrix exponential; see Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review, 1978. This routine uses Ward’s diagonal Padé approximation method with three step preconditioning (SIAM Journal on Numerical Analysis, 1977). Diagonal Padé approximations are rational polynomials of matrices

```     -1
D (A)   N (A)
```

whose Taylor series matches the first `2q+1` terms of the Taylor series above; direct evaluation of the Taylor series (with the same preconditioning steps) may be desirable in lieu of the Padé approximation when `Dq(A)` is ill-conditioned.

Function File: s = logm (A)
Function File: s = logm (A, opt_iters)
Function File: [s, iters] = logm (…)

Compute the matrix logarithm of the square matrix A.

The implementation utilizes a Padé approximant and the identity

```logm (A) = 2^k * logm (A^(1 / 2^k))
```

The optional input opt_iters is the maximum number of square roots to compute and defaults to 100.

The optional output iters is the number of square roots actually computed.

Built-in Function: s = sqrtm (A)
Built-in Function: [s, error_estimate] = sqrtm (A)

Compute the matrix square root of the square matrix A.

Ref: N.J. Higham. A New sqrtm for MATLAB. Numerical Analysis Report No. 336, Manchester Centre for Computational Mathematics, Manchester, England, January 1999.

Built-in Function: kron (A, B)
Built-in Function: kron (A1, A2, …)

Form the Kronecker product of two or more matrices.

This is defined block by block as

```x = [ a(i,j)*b ]
```

For example:

```kron (1:4, ones (3, 1))
⇒  1  2  3  4
1  2  3  4
1  2  3  4
```

If there are more than two input arguments A1, A2, …, An the Kronecker product is computed as

```kron (kron (A1, A2), …, An)
```

Since the Kronecker product is associative, this is well-defined.

Built-in Function: blkmm (A, B)

Compute products of matrix blocks.

The blocks are given as 2-dimensional subarrays of the arrays A, B. The size of A must have the form `[m,k,…]` and size of B must be `[k,n,…]`. The result is then of size `[m,n,…]` and is computed as follows:

```for i = 1:prod (size (A)(3:end))
C(:,:,i) = A(:,:,i) * B(:,:,i)
endfor
```
Built-in Function: X = syl (A, B, C)

Solve the Sylvester equation

```A X + X B = C
```

using standard LAPACK subroutines.

For example:

```sylvester ([1, 2; 3, 4], [5, 6; 7, 8], [9, 10; 11, 12])
⇒ [ 0.50000, 0.66667; 0.66667, 0.50000 ]
```

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