funm

Matrix Cube Root

Find matrix B, such that B3 = A, where A is a 3-by-3 identity matrix.

To solve B3 = A, compute the cube root of the matrix 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 B3 = A.

B^3

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

Matrix Lambert W Function

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 (W0 branch) in a matrix sense, callfunm 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·eW0 within the specified accuracy.

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-1 of the Lambert W function.

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

Find the W-1 branch for the matrix 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·eWm1 within the specified accuracy.

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]

Matrix Exponential, Logarithm, and Square Root

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)]