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Quantitative Analysis
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Numerical Analysis
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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.

Gauss-Hermite Integration.


he following is an extremely efficient integration formula:

MATH (Gauss-Hermite Intergration)
MATH Note that one can do the change of function MATH to obtain more generic looking result.

The below values of $w_{i},x_{i}$ are taken from [Abramowitz] , pages 890 and 924: MATH MATH MATH What follows next is a fragment of theory of orthogonal polynomials that leads to the formula ( Gauss-Hermite Integration ). The proposition ( Gaussian quadrature rule ) provides the justification. There are several sections after ( Gaussian quadrature rule ) included for their importance for other applications within these Notes. The reference is [Gautschi] .




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.

Notation. Index. Contents.


















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