Lagrange polynomial

### 17.1.1 Lagrange polynomial

The lagrange command finds the Lagrange polynomial which interpolates given data.

- lagrange takes two mandatory arguments and one optional argument:
  - l1 and l2, two lists of the same size. These can be given as a matrix with two rows.

    The first list (resp. row) corresponds to the abscissa values xk (k=1,…,n), and the second list (resp. row) corresponds to ordinate values yk (k=1,…,n).

  - Optionally x, the name of a variable (by default x).
- lagrange(l1,l2 ⟨,x⟩) returns a polynomial expression P(x) of degree n−1 such that P(xi)=yi.

You can use the interp command as a synonym for lagrange.

#### Examples

lagrange([[1,3],[0,1]])

or:

lagrange([1,3],[0,1])

x−1
2

since x−1/2=0 for x=1 and x−1/2=1 for x=3.

factor(lagrange([1,2,3,4],[1,a,-2a,0],t))

⎛ ⎝ | t−4 | ⎞ ⎠ | ⎛ ⎝ | 9 a t2−30 a t+21 a−t2+5 t−6 | ⎞ ⎠
6

##### Remark.

An attempted function definition such as f:=lagrange([1,2],[3,4],y) does not return a function but an expression with respect to y. To define f as a function, input:

f:=unapply(lagrange([1,2],[3,4],x),x)

Avoid f(x):=lagrange([1,2],[3,4],x) since then the Lagrange polynomial would be computed each time f is called (indeed in a function definition, the second member of the assignment is not evaluated). Note also that g(x):=lagrange([1,2],[3,4]) would not work since the default argument of lagrange would be global, hence not the same as the local variable used for the definition of g.