# polyeval¶

## Purpose¶

Evaluates polynomials. Can either be one or more scalar polynomials or a single matrix polynomial.

## Format¶

y = polyeval(x, coefs)
Parameters
• x (1xK vector or NxN matrix.) – can either represent K separate scalar values at which to evaluate the (scalar) polynomial(s), or it can represent a single NxN matrix.

• c ((P+1)xK or (P+1)x1 matrix) – coefficients of polynomials to evaluate. If x is 1xK, then c must be (P+1)xK. If x is NxN, c must be (P+1)x1. That is, if x is a matrix, it can only be evaluated at a single set of coefficients.

Returns

y (Kx1 vector or NxN matrix) –

Kx1 vector (if c is (P+1)xK) or NxN matrix (if c is (P+1)x1 and x is NxN):

$y =( c[1, .]*x^p + c[2, .]*x^{(p-1)} + ... + c[p+1, .] )';$

## Examples¶

### Scalar example 1¶

// Evaluate 2^4 + 2^3 + 2^1 + 2^0
x = 2;
coefs = { 1, 1, 0, 1, 1 };
y = polyeval(x, coefs);


The result is 27. Note that this is the decimal value of the binary number 11011.

### Scalar example 2¶

// Evaluate 7*2^3 + 2^1 + 2^0
x = 2;
coefs = { 7, 0, 1, 1 };
y = polyeval(x, coefs);


The result is 59.

### Matrix example 1¶

// Evaluate A*A*A + 2*A
A = { 2 6,
4 8 };
coefs = { 1, 0, 2, 0 };
A_3 = polyeval(A, coefs);


The above code will set A_3 equal to:

300    660
440    960


### Matrix example 2¶

// Evaluate A*A*A*A
A = { 1.2 3.1,
1.7 0.8 };
coefs = 1|zeros(4,1);
y = polyeval(A, coefs);


You can raise a matrix to the n’th power with the command:

A_n = polyeval(A, 1|zeros(n, 1));


(e.g: A*A*A*A*…*A).

## Remarks¶

In both the scalar and the matrix case, Horner’s rule is used to do the evaluation. In the scalar case, the function func:recsercp is called (this implements an elaboration of Horner’s rule).

poly.src