naginterfaces.library.fit.dim1_​cheb_​deriv

naginterfaces.library.fit.dim1_cheb_deriv(n, xmin, xmax, a, ia1, iadif1)

dim1_cheb_deriv determines the coefficients in the Chebyshev series representation of the derivative of a polynomial given in Chebyshev series form.

For full information please refer to the NAG Library document for e02ah

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/e02/e02ahf.html

Parameters

nint

$n$, the degree of the given polynomial $p\left(x\right)$.

xminfloat

The lower and upper end points respectively of the interval $[x_{min},x_{max}]$. The Chebyshev series representation is in terms of the normalized variable $\overline{x}$, where

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

xmaxfloat

The lower and upper end points respectively of the interval $[x_{min},x_{max}]$. The Chebyshev series representation is in terms of the normalized variable $\overline{x}$, where

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

afloat, array-like, shape $(1+n\times ia1)$

The Chebyshev coefficients of the polynomial $p\left(x\right)$. Specifically, element $\text{i}\times ia1$ of $a$ must contain the coefficient $a_{\text{i}}$, for $\text{i}=0,1,\dots ,n$. Only these $n+1$ elements will be accessed.

Unchanged on exit, but see $adif$, below.

ia1int

The index increment of $a$. Most frequently the Chebyshev coefficients are stored in adjacent elements of $a$, and $ia1$ must be set to $1$. However, if for example, they are stored in $a\left[0\right],a\left[3\right],a\left[6\right],\dots$, the value of $ia1$ must be $3$. See also Further Comments.

iadif1int

The index increment of $adif$. Most frequently the Chebyshev coefficients are required in adjacent elements of $adif$, and $iadif1$ must be set to $1$. However, if, for example, they are to be stored in $adif\left[0\right],adif\left[3\right],adif\left[6\right],\dots$, the value of $iadif1$ must be $3$. See Further Comments.

Returns

patm1float

The value of $p\left(x_{min}\right)$. If this value is passed to the integration function dim1_cheb_integ() with the coefficients of $q\left(x\right)$, the original polynomial $p\left(x\right)$ is recovered, including its constant coefficient.

adiffloat, ndarray, shape $(1+n\times iadif1)$

The Chebyshev coefficients of the derived polynomial $q\left(x\right)$. (The differentiation is with respect to the variable $x$.) Specifically, element $\text{i}\times iadif1$ of $adif$ contains the coefficient ${\overline{a}}_{\text{i}}$, for $\text{i}=0,1,\dots ,n-1$. Additionally, element $n\times iadif1$ is set to zero. A call of the function may have the array name $adif$ the same as $a$, provided that note is taken of the order in which elements are overwritten, when choosing the starting elements and increments $ia1$ and $iadif1$, i.e., the coefficients $a_0,a_1,\dots ,a_{i-1}$ must be intact after coefficient ${\overline{a}}_i$ is stored. In particular, it is possible to overwrite the $a_i$ completely by having $ia1=iadif1$, and the actual arrays for $a$ and $adif$ identical.

Raises

NagValueError

(errno $1$)

On entry, $\text{ladif}=⟨value⟩$, $n=⟨value⟩$ and $iadif1=⟨value⟩$.

Constraint: $\text{ladif}>n\times iadif1$.

(errno $1$)

On entry, $iadif1=⟨value⟩$.

Constraint: $iadif1\ge 1$.

(errno $1$)

On entry, $\text{la}=⟨value⟩$, $n=⟨value⟩$ and $ia1=⟨value⟩$.

Constraint: $\text{la}>n\times ia1$.

(errno $1$)

On entry, $ia1=⟨value⟩$.

Constraint: $ia1\ge 1$.

(errno $1$)

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

Constraint: $xmax>xmin$.

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 0$.

Notes

dim1_cheb_deriv forms the polynomial which is the derivative of a given polynomial. Both the original polynomial and its derivative are represented in Chebyshev series form. Given the coefficients $a_{\text{i}}$, for $\text{i}=0,1,\dots ,n$, of a polynomial $p\left(x\right)$ of degree $n$, where

$$p\left(x\right)=\frac{1}{2}{a}_0+a_1{T}_1\left(\overline{x}\right)+\cdots +a_n{T}_n\left(\overline{x}\right)$$

the function returns the coefficients ${\overline{a}}_{\text{i}}$, for $\text{i}=0,1,\dots ,n-1$, of the polynomial $q\left(x\right)$ of degree $n-1$, where

$$q\left(x\right)=\frac{dp\left(x\right)}{dx}=\frac{1}{2}{\overline{a}}_0+{\overline{a}}_1{T}_1\left(\overline{x}\right)+\cdots +{\overline{a}}_{n-1}{T}_{n-1}\left(\overline{x}\right)\text{.}$$

Here $T_j\left(\overline{x}\right)$ denotes the Chebyshev polynomial of the first kind of degree $j$ with argument $\overline{x}$. It is assumed that the normalized variable $\overline{x}$ in the interval $[-1,+1]$ was obtained from your original variable $x$ in the interval $[x_{min},x_{max}]$ by the linear transformation

$$\overline{x}=\frac{2x-(x_{max}+x_{min})}{x_{max}-x_{min}}$$

and that you require the derivative to be with respect to the variable $x$. If the derivative with respect to $\overline{x}$ is required, set $x_{max}=1$ and $x_{min}=-1$.

Values of the derivative can subsequently be computed, from the coefficients obtained, by using dim1_cheb_eval2().

The method employed is that of Chebyshev series (see Module 8 of Modern Computing Methods (1961)), modified to obtain the derivative with respect to $x$. Initially setting ${\overline{a}}_{n+1}={\overline{a}}_n=0$, the function forms successively

$${\overline{a}}_{i-1}={\overline{a}}_{i+1}+\frac{2}{x_{max}-x_{min}}2i{a}_i\text{,}i=n,n-1,\dots ,1\text{.}$$

References

Modern Computing Methods, 1961, Chebyshev-series, NPL Notes on Applied Science (16), (2nd Edition), HMSO