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1: 3.5 Quadrature

… ►

§3.5(v) Gauss Quadrature

… ►In adaptive algorithms the evaluation of the nodes and weights may cause difficulties, unless exact values are known. … ►The nodes

x_{k}

and weights

w_{k}

are known explicitly: … ►The complex Gauss nodes

\zeta_{k}

have positive real part for all

s>0

. … ►Extensive tables of quadrature nodes and weights can be found in Krylov and Skoblya (1985). …

Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ .

…

3: 3.3 Interpolation

… ►

§3.3(i) Lagrange Interpolation

►The nodes or abscissas

z_{k}

are real or complex; function values are

f_{k}=f(z_{k})

. … ►If

f

x

(

=z

), and the nodes

x_{k}

are real, and

f^{(n+1)}

is continuous on the smallest closed interval

I

containing

x,x_{0},x_{1},\dots,x_{n}

, then the error can be expressed … ►

§3.3(ii) Lagrange Interpolation with Equally-Spaced Nodes

… ►where the nodes

x_{k}=x_{0}+kh

(

h>0

) and function

f

are real, …

4: 3.4 Differentiation

… ►

§3.4(i) Equally-Spaced Nodes

… ►For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). …

5: 18.38 Mathematical Applications

… ►If the nodes in a quadrature formula with a positive weight function are chosen to be the zeros of the

n

th degree OP with the same weight function, and the interval of orthogonality is the same as the integration range, then the weights in the quadrature formula can be chosen in such a way that the formula is exact for all polynomials of degree not exceeding

2n-1

. …

6: 18.39 Applications in the Physical Sciences

… ►Namely the

k

th eigenfunction, listed in order of increasing eigenvalues, starting at

k=0

, has precisely

k

nodes, as real zeros of wave-functions away from boundaries are often referred to. …,

k=1

, there might be 3 nodes, rather than Sturm’s 1. …Thus the two missing quantum numbers corresponding to EOP’s of order

1

and

2

of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. …

7: 1.13 Differential Equations

… ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues,

\lambda

; (ii) the corresponding (real) eigenfunctions,

u(x)

and

w(t)

, have the same number of zeros, also called nodes, for

t\in(0,c)

as for

x\in(a,b)

; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

8: 3.11 Approximation Techniques

… ►Given

n+1

distinct points

x_{k}

in the real interval

[a,b]

, with (

a=

)

x_{0}<x_{1}<\cdots<x_{n-1}<x_{n}

(

=b

), on each subinterval

[x_{k},x_{k+1}]

k=0,1,\ldots,n-1

, a low-degree polynomial is defined with coefficients determined by, for example, values

f_{k}

and

f_{k}^{\prime}

of a function

f

and its derivative at the nodes

x_{k}

and

x_{k+1}

. …