naginterfaces.library.fit.dim1_​minimax_​polynomial

naginterfaces.library.fit.dim1_minimax_polynomial(x, y, m)

dim1_minimax_polynomial calculates a minimax polynomial fit to a set of data points.

For full information please refer to the NAG Library document for e02al

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/e02/e02alf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The values of the $x$ coordinates, $x_i$, for $\text{i}=1,2,\dots ,n$.

yfloat, array-like, shape $\left(n\right)$

The values of the $y$ coordinates, $y_i$, for $\text{i}=1,2,\dots ,n$.

mint

$m$, where $m$ is the degree of the polynomial to be found.

Returns

afloat, ndarray, shape $(m+1)$

The coefficients $a_i$ of the minimax polynomial, for $\text{i}=0,1,\dots ,m$.

reffloat

The final reference deviation, i.e., the maximum deviation of the computed polynomial evaluated at $x_{\text{i}}$ from the reference values $y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$. $ref$ may return a negative value which indicates that the algorithm started to cycle due to round-off errors.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 0$.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m<100$.

(errno $2$)

On entry, $m=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $m<n-1$.

(errno $3$)

On entry, $\text{i}=⟨value⟩$, $x\left[\text{i}\right]=⟨value⟩$ and $x[\text{i}-1]=⟨value⟩$.

Constraint: $x\left[\text{i}\right]>x[\text{i}-1]$.

Notes

Given a set of data points $(x_i,y_i)$, for $\text{i}=1,2,\dots ,n$, dim1_minimax_polynomial uses the exchange algorithm to compute an $m$th-degree polynomial

$$P\left(x\right)=a_0+a_{1}x+a_2{x}^2+\cdots +a_m{x}^m$$

such that ${max}_i|P\left(x_i\right)-y_i|$ is a minimum.

The function also returns a number whose absolute value is the final reference deviation (see Parameters). The function is an adaptation of Boothroyd (1967).

References

Boothroyd, J B, 1967, Algorithm 318, Comm. ACM (10), 801

Stieffel, E, 1959, Numerical methods of Tchebycheff approximation, On Numerical Approximation, (ed R E Langer), 217–232, University of Wisconsin Press