General matrix function

computes
the function `F`

= funm(`A`

,`f`

)`f`

(`A`

) for the
square matrix `A`

. For details, see Matrix Function.

Find matrix `B`

, such that `B`

, where ^{3} =
A`A`

is a 3-by-3 identity matrix.

To solve `B`

,
compute the cube root of the matrix ^{3} = A`A`

using the `funm`

function.
Create the symbolic function `f(x) = x^(1/3)`

and
use it as the second argument for `funm`

. The cube
root of an identity matrix is the identity matrix itself.

A = sym(eye(3)) syms f(x) f(x) = x^(1/3); B = funm(A,f)

A = [ 1, 0, 0] [ 0, 1, 0] [ 0, 0, 1] B = [ 1, 0, 0] [ 0, 1, 0] [ 0, 0, 1]

Replace one of the `0`

elements of matrix `A`

with `1`

and
compute the matrix cube root again.

A(1,2) = 1 B = funm(A,f)

A = [ 1, 1, 0] [ 0, 1, 0] [ 0, 0, 1] B = [ 1, 1/3, 0] [ 0, 1, 0] [ 0, 0, 1]

Now, compute the cube root of the upper triangular matrix.

A(1:2,2:3) = 1 B = funm(A,f)

A = [ 1, 1, 1] [ 0, 1, 1] [ 0, 0, 1] B = [ 1, 1/3, 2/9] [ 0, 1, 1/3] [ 0, 0, 1]

Verify that `B`

.^{3} = A

B^3

ans = [ 1, 1, 1] [ 0, 1, 1] [ 0, 0, 1]

Find the matrix Lambert W function.

First, create a 3-by-3 matrix `A`

using variable-precision
arithmetic with five digit accuracy. In this example, using variable-precision
arithmetic instead of exact symbolic numbers lets you speed up computations
and decrease memory usage. Using only five digits helps the result
to fit on screen.

savedefault = digits(5); A = vpa(magic(3))

A = [ 8.0, 1.0, 6.0] [ 3.0, 5.0, 7.0] [ 4.0, 9.0, 2.0]

Create the symbolic function `f(x) = lambertw(x)`

.

syms f(x) f(x) = lambertw(x);

To find the Lambert W function (`W`

branch)
in a matrix sense, call_{0}`funm`

using `f(x)`

as
its second argument.

W0 = funm(A,f)

W0 = [ 1.5335 + 0.053465i, 0.11432 + 0.47579i, 0.36208 - 0.52925i] [ 0.21343 + 0.073771i, 1.3849 + 0.65649i, 0.41164 - 0.73026i] [ 0.26298 - 0.12724i, 0.51074 - 1.1323i, 1.2362 + 1.2595i]

Verify that this result is a solution of the matrix equation ```
A
= W0·e
```

within the specified
accuracy.^{W0}

W0*expm(W0)

ans = [ 8.0, 1.0 - 5.6843e-13i, 6.0 + 1.1369e-13i] [ 3.0 - 2.2737e-13i, 5.0 - 2.8422e-14i, 7.0 - 4.1211e-13i] [ 4.0 - 2.2737e-13i, 9.0 - 9.9476e-14i, 2.0 + 1.4779e-12i]

Now, create the symbolic function `f(x)`

representing
the branch `W`

of the Lambert
W function._{-1}

f(x) = lambertw(-1,x);

Find the `W`

branch
for the matrix _{-1}`A`

.

Wm1 = funm(A,f)

Wm1 = [ 0.40925 - 4.7154i, 0.54204 + 0.5947i, 0.13764 - 0.80906i] [ 0.38028 + 0.033194i, 0.65189 - 3.8732i, 0.056763 - 1.0898i] [ 0.2994 - 0.24756i, - 0.105 - 1.6513i, 0.89453 - 3.0309i]

Verify that this result is the solution of the matrix equation ```
A
= Wm1·e
```

within the specified
accuracy.^{Wm1}

Wm1*expm(Wm1)

ans = [ 8.0 - 8.3844e-13i, 1.0 - 3.979e-13i, 6.0 - 9.0949e-13i] [ 3.0 - 9.6634e-13i, 5.0 + 1.684e-12i, 7.0 + 4.5475e-13i] [ 4.0 - 1.3642e-12i, 9.0 + 1.6698e-12i, 2.0 + 1.7053e-13i]

You can use `funm`

with
appropriate second arguments to find matrix exponential, logarithm,
and square root. However, the more efficient approach is to use the
functions `expm`

, `logm`

, and `sqrtm`

for
this task.

Create this square matrix and find its exponential, logarithm, and square root.

syms x A = [1 -1; 0 x] expA = expm(A) logA = logm(A) sqrtA = sqrtm(A)

A = [ 1, -1] [ 0, x] expA = [ exp(1), (exp(1) - exp(x))/(x - 1)] [ 0, exp(x)] logA = [ 0, -log(x)/(x - 1)] [ 0, log(x)] sqrtA = [ 1, 1/(x - 1) - x^(1/2)/(x - 1)] [ 0, x^(1/2)]

Find the matrix exponential, logarithm, and square root of `A`

using `funm`

.
Use the symbolic expressions `exp(x)`

, `log(x)`

,
and `sqrt(x)`

as the second argument of `funm`

.
The results are identical.

expA = funm(A,exp(x)) logA = funm(A,log(x)) sqrtA = funm(A,sqrt(x))

expA = [ exp(1), exp(1)/(x - 1) - exp(x)/(x - 1)] [ 0, exp(x)] logA = [ 0, -log(x)/(x - 1)] [ 0, log(x)] sqrtA = [ 1, 1/(x - 1) - x^(1/2)/(x - 1)] [ 0, x^(1/2)]

For compatibility with the MATLAB

^{®}`funm`

function,`funm`

accepts the following arguments:Function handles such as

`@exp`

and`@sin`

, as its second input argument.The

`options`

input argument, such as`funm(A,f,options)`

.Additional input arguments of the function

`f`

, such as`funm(A,f,options,p1,p2,...)`

The

`exitflag`

output argument, such as`[F,exitflag] = funm(A,f)`

. Here,`exitflag`

is`1`

only if the`funm`

function call errors, for example, if it encounters a division by zero. Otherwise,`exitflag`

is`0`

.

For more details and a list of all acceptable function handles, see the MATLAB

`funm`

function.If the input matrix

`A`

is numeric (not a symbolic object) and the second argument`f`

is a function handle, then the`funm`

call invokes the MATLAB`funm`

function.