Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
A. Finite difference basics.
B. One dimensional heat equation.
a. Finite difference schemes for heat equation.
b. Stability of one-dim heat equation schemes.
c. Remark on stability of financial problems.
d. Lagrangian coordinate technique.
e. Factorization procedure for heat equation.
C. Two dimensional heat equation.
D. General techniques for reduction of dimensionality.
E. Time dependent case.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Factorization procedure for heat equation.

here is a way to solve a three-diagonal finite difference scheme with an $N-$ linear number of algebraic operations. Suppose some finite difference scheme takes the form

MATH (Factorization1)
MATH MATH with the properties
MATH (Factorization3)
MATH (Factorization4)
We look for a recursive relationship
MATH (Factorization2)
with some coefficients MATH that we determine through the substitution into the scheme ( Factorization1 ): MATH We express all the $y_{\cdot}$ through the $y_{i+1}$ using the relationship ( Factorization2 ): MATH MATH Hence, to satisfy the ( Factorization1 ) we must have $\{...\}=0$ and MATH Such conditions lead to the recursive relationships
MATH (Factorization5)
MATH (Factorization6)
Under conditions ( Factorization3 ),( Factorization4 ) we have MATH MATH for all $i$ . Therefore, we start with MATH and produce MATH $i=1,...,N$ according to the ( Factorization5 ),( Factorization6 ). At the final step of the iteration we use the boundary condition MATH MATH Afterward, with MATH $i=1,...,N$ already known we compute $y_{i},$ with recursion in opposite direction $i=N-1,...$ according to the ( Factorization2 ).

Notation. Index. Contents.

Copyright 2007