Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
A. Gram-Schmidt orthogonalization.
B. Definition and existence of orthogonal polynomials.
C. Three-term recurrence relation for orthogonal polynomials.
D. Orthogonal polynomials and quadrature rules.
E. Extremal properties of orthogonal polynomials.
F. Chebyshev polynomials.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Orthogonal polynomials and quadrature rules.


(Quadrature rule) Let MATH and MATH are arbitrary numbers and $d\lambda$ is a measure on $\QTR{cal}{R}$ . We introduce the notations MATH The "quadrature rule"

MATH (Quadrature rule 1)
is said to have "degree of exactness" $d$ iff MATH The "precise degree of exactness" is the maximal degree of exactness.

The quadrature rule with degree of exactness $n-1$ is called "interpolatory".


(Lagrange interpolation formula) For a set MATH we define MATH according to the relationship

MATH (Lagrange interpolation formula 1)

Note that MATH hence MATH Therefore MATH Indeed, MATH has the form MATH . It is defined by values in $n$ distinct points.


(Coefficients of quadrature rule) For a given set MATH there exist MATH such that the quadrature rule ( Quadrature rule 1 ) is interpolatory.


Integrate the formula ( Lagrange interpolation formula 1 ).


(Higher degree of exactness) Given an integer $k,~0\leq k\leq n$ , the formula ( Quadrature rule 1 ) has degree of exactness $d=n-1+k$ iff both of the following conditions are satisfied:

1. The formula ( Quadrature rule 1 ) is interpolatory.

2. The polynomial MATH satisfies MATH


(Necessity) The claim (1) follows immediately by the definition ( Quadrature rule ).

The claim (2) follows by substitution $f=$ $\omega_{n}p$ into the formula ( Quadrature rule 1 ): MATH The term MATH is zero because MATH and term MATH is zero because MATH .

(Sufficiency) We assume (1),(2) and proceed to show that MATH We divide MATH by $\omega_{n}$ and represent the result as MATH where MATH and MATH . Then MATH The term MATH is zero by (2). Thus MATH and MATH because MATH and (1). We conclude MATH since MATH .


(Gaussian quadrature rule) Let MATH be the polynomials related to measure $d\lambda$ as in the definition ( Orthogonal polynomials ) and the inner product MATH is positive definite. Then the formula ( Quadrature rule 1 ) has degree of exactness $2n-1$ iff $\omega_{n}=\pi_{n}$ and $\tau_{s}$ are zeros of $\pi_{n}$ .


We apply the proposition ( Higher degree of exactness ). The condition (2) requires that $\omega_{n}=\pi_{n}$ and $\tau_{s}$ are zeros of $\pi_{n}$ by definition of $\omega_{n}$ . The numbers MATH are all real by the proposition ( Zeros of orthogonal polynomials ).

The condition (1) becomes evident after noting similarity in structure of the polynomial $\omega_{n}$ and the polynomial $L_{\cdot}$ of the definition ( Lagrange interpolation formula ).

The coefficients MATH may be found from the proposition ( Coefficients of quadrature rule ).

Notation. Index. Contents.

Copyright 2007