naginterfaces.library.fit.dim1_​cheb_​integ

naginterfaces.library.fit.dim1_cheb_integ(n, xmin, xmax, a, ia1, qatm1, iaint1)

dim1_cheb_integ determines the coefficients in the Chebyshev series representation of the indefinite integral of a polynomial given in Chebyshev series form.

For full information please refer to the NAG Library document for e02aj

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/e02/e02ajf.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 $aintc$, 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.

qatm1float

The value that the integrated polynomial is required to have at the lower end point of its interval of definition, i.e., at $\overline{x}=-1$ which corresponds to $x=x_{min}$. Thus, $qatm1$ is a constant of integration and will normally be set to zero by you.

iaint1int

The index increment of $aintc$. Most frequently the Chebyshev coefficients are required in adjacent elements of $aintc$, and $iaint1$ must be set to $1$. However, if, for example, they are to be stored in $aintc\left[0\right],aintc\left[3\right],aintc\left[6\right],\dots$, the value of $iaint1$ must be $3$. See also Further Comments.

Returns

aintcfloat, ndarray, shape $(1+(n+1)\times iaint1)$

The Chebyshev coefficients of the integral $q\left(x\right)$. (The integration is with respect to the variable $x$, and the constant coefficient is chosen so that $q\left(x_{min}\right)$ equals $qatm1$). Specifically, element $i\times iaint1$ of $aintc$ contains the coefficient $a_{\text{i}}^{\prime }$, for $\text{i}=0,1,\dots ,n+1$. A call of the function may have the array name $aintc$ the same as $a$, provided that note is taken of the order in which elements are overwritten when choosing starting elements and increments $ia1$ and $iaint1$: i.e., the coefficients, $a_0,a_1,\dots ,a_{i-2}$ must be intact after coefficient $a_i^{\prime }$ is stored. In particular it is possible to overwrite the $a_i$ entirely by having $ia1=iaint1$, and the actual array for $a$ and $aintc$ identical.

Raises

NagValueError

(errno $1$)

On entry, $\text{laint}=⟨value⟩$, $n=⟨value⟩$ and $iaint1=⟨value⟩$.

Constraint: $\text{laint}>(n+1)\times iaint1$.

(errno $1$)

On entry, $iaint1=⟨value⟩$.

Constraint: $iaint1\ge 1$.

(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_integ forms the polynomial which is the indefinite integral of a given polynomial. Both the original polynomial and its integral are represented in Chebyshev series form. If supplied with the coefficients $a_i$, for $\text{i}=0,1,\dots ,n$, of a polynomial $p\left(x\right)$ of degree $n$, where

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

the function returns the coefficients $a_i^{\prime }$, for $\text{i}=0,1,\dots ,n+1$, of the polynomial $q\left(x\right)$ of degree $n+1$, where

$$q\left(x\right)=\frac{1}{2}{a}_0^{\prime }+a_1^{\prime }{T}_1\left(\overline{x}\right)+\cdots +a_{n+1}^{\prime }{T}_{n+1}\left(\overline{x}\right)\text{,}$$

and

$$q\left(x\right)=\int p\left(x\right)dx\text{.}$$

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

Values of the integral 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 for integrating with respect to $x$. Initially taking $a_{n+1}=a_{n+2}=0$, the function forms successively

$$a_i^{\prime }=\frac{a_{i-1}-a_{i+1}}{2i}\times \frac{x_{max}-x_{min}}{2}\text{,}i=n+1,n,\dots ,1\text{.}$$

The constant coefficient $a_0^{\prime }$ is chosen so that $q\left(x\right)$ is equal to a specified value, $qatm1$, at the lower end point of the interval on which it is defined, i.e., $\overline{x}=-1$, which corresponds to $x=x_{min}$.

References

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