naginterfaces.library.fit.dim2_​cheb_​eval

naginterfaces.library.fit.dim2_cheb_eval(mfirst, k, l, x, xmin, xmax, y, ymin, ymax, a)

dim2_cheb_eval evaluates a bivariate polynomial from the rectangular array of coefficients in its double Chebyshev series representation.

For full information please refer to the NAG Library document for e02cb

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/e02/e02cbf.html

Parameters

mfirstint

The index of the first and last $x$ value in the array $x$ at which the evaluation is required respectively (see Further Comments).

kint

The degree $k$ of $x$ and $l$ of $y$, respectively, in the polynomial.

lint

The degree $k$ of $x$ and $l$ of $y$, respectively, in the polynomial.

xfloat, array-like, shape $\left(\text{mlast}\right)$

$x[\text{i}-1]$, for $\text{i}=mfirst,\dots ,\text{mlast}$, must contain the $x$ values at which the evaluation is required.

xminfloat

The lower and upper ends, $x_{min}$ and $x_{max}$, of the range of the variable $x$ (see Notes).

The values of $xmin$ and $xmax$ may depend on the value of $y$ (e.g., when the polynomial has been derived using dim2_cheb_lines()).

xmaxfloat

The lower and upper ends, $x_{min}$ and $x_{max}$, of the range of the variable $x$ (see Notes).

The values of $xmin$ and $xmax$ may depend on the value of $y$ (e.g., when the polynomial has been derived using dim2_cheb_lines()).

yfloat

The value of the $y$ coordinate of all the points at which the evaluation is required.

yminfloat

The lower and upper ends, $y_{min}$ and $y_{max}$, of the range of the variable $y$ (see Notes).

ymaxfloat

The lower and upper ends, $y_{min}$ and $y_{max}$, of the range of the variable $y$ (see Notes).

afloat, array-like, shape $((k+1)\times (l+1))$

The Chebyshev coefficients of the polynomial. The coefficient $a_{ij}$ defined according to the standard convention (see Notes) must be in $a[i\times (l+1)+j]$.

Returns

fffloat, ndarray, shape $\left(\text{mlast}\right)$

$ff[\text{i}-1]$ gives the value of the polynomial at the point $(x_{\text{i}},y)$, for $\text{i}=mfirst,\dots ,\text{mlast}$.

Raises

NagValueError

(errno $1$)

On entry, $\text{na}=⟨value⟩$, $k=⟨value⟩$ and $l=⟨value⟩$.

Constraint: $\text{na}\ge (k+1)\times (l+1)$.

(errno $1$)

On entry, $l=⟨value⟩$.

Constraint: $l\ge 0$.

(errno $1$)

On entry, $k=⟨value⟩$.

Constraint: $k\ge 0$.

(errno $1$)

On entry, $mfirst=⟨value⟩$ and $\text{mlast}=⟨value⟩$.

Constraint: $mfirst\le \text{mlast}$.

(errno $2$)

On entry, $y=⟨value⟩$ and $ymax=⟨value⟩$.

Constraint: $y\le ymax$.

(errno $2$)

On entry, $y=⟨value⟩$ and $ymin=⟨value⟩$.

Constraint: $y\ge ymin$.

(errno $2$)

On entry, $ymin=⟨value⟩$ and $ymax=⟨value⟩$.

Constraint: $ymin<ymax$.

(errno $3$)

On entry, $\text{I}=⟨value⟩$, $x[\text{I}-1]=⟨value⟩$ and $xmax=⟨value⟩$.

Constraint: $x[\text{I}-1]\le xmax$.

(errno $3$)

On entry, $\text{I}=⟨value⟩$, $x[\text{I}-1]=⟨value⟩$ and $xmin=⟨value⟩$.

Constraint: $x[\text{I}-1]\ge xmin$.

(errno $3$)

On entry, $xmin=⟨value⟩$ and $xmax=⟨value⟩$.

Constraint: $xmin<xmax$.

(errno $4$)

Unexpected internal failure when evaluating the polynomial.

Notes

This function evaluates a bivariate polynomial (represented in double Chebyshev form) of degree $k$ in one variable, $\overline{x}$, and degree $l$ in the other, $\overline{y}$. The range of both variables is $-1$ to $+1$. However, these normalized variables will usually have been derived (as when the polynomial has been computed by dim2_cheb_lines(), for example) from your original variables $x$ and $y$ by the transformations

$$\overline{x}=\frac{2x-(x_{max}+x_{min})}{(x_{max}-x_{min})}\text{and}\overline{y}=\frac{2y-(y_{max}+y_{min})}{(y_{max}-y_{min})}\text{.}$$

(Here $x_{min}$ and $x_{max}$ are the ends of the range of $x$ which has been transformed to the range $-1$ to $+1$ of $\overline{x}$. $y_{min}$ and $y_{max}$ are correspondingly for $y$. See Further Comments). For this reason, the function has been designed to accept values of $x$ and $y$ rather than $\overline{x}$ and $\overline{y}$, and so requires values of $x_{min}$, etc. to be supplied by you. In fact, for the sake of efficiency in appropriate cases, the function evaluates the polynomial for a sequence of values of $x$, all associated with the same value of $y$.

The double Chebyshev series can be written as

$$\sum _{i=0}^k\sum _{j=0}^l{a}_{ij}{T}_i\left(\overline{x}\right)T_j\left(\overline{y}\right)\text{,}$$

where $T_i\left(\overline{x}\right)$ is the Chebyshev polynomial of the first kind of degree $i$ and argument $\overline{x}$, and $T_j\left(\overline{y}\right)$ is similarly defined. However the standard convention, followed in this function, is that coefficients in the above expression which have either $i$ or $j$ zero are written $\frac{1}{2}{a}_{ij}$, instead of simply $a_{ij}$, and the coefficient with both $i$ and $j$ zero is written $\frac{1}{4}{a}_{0,0}$.

The function first forms $c_i={\sum }_{j=0}^l{a}_{ij}{T}_j\left(\overline{y}\right)$, with $a_{i,0}$ replaced by $\frac{1}{2}{a}_{i,0}$, for each of $i=0,1,\dots ,k$. The value of the double series is then obtained for each value of $x$, by summing $c_i\times T_i\left(\overline{x}\right)$, with $c_0$ replaced by $\frac{1}{2}{c}_0$, over $i=0,1,\dots ,k$. The Clenshaw three term recurrence (see Clenshaw (1955)) with modifications due to Reinsch and Gentleman (1969) is used to form the sums.

References

Clenshaw, C W, 1955, A note on the summation of Chebyshev series, Math. Tables Aids Comput. (9), 118–120

Gentleman, W M, 1969, An error analysis of Goertzel’s (Watt’s) method for computing Fourier coefficients, Comput. J. (12), 160–165