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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.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
A. Quadratic form minimum.
B. Method of steepest descent.
C. Method of conjugate directions.
D. Method of conjugate gradients.
E. Convergence analysis of conjugate gradient method.
F. Preconditioning.
G. Recursive calculation.
H. Parallel subspace preconditioner.
2. Wavelet Analysis.
3. Finite element method.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Method of steepest descent.

e consider the problem ( Quadratic form minimum ). We propose to construct the sequence MATH as follows.

Start with any $x_{0}$ .

Suppose we are after step $k$ so that $x_{k}$ is already calculated. Evaluate the direction of decay MATH and set MATH We chose a $\tau_{k}$ to minimize MATH MATH and then set MATH We perform the following calculations to complete the recipe: MATH Therefore, MATH

MATH (Orthogonality of residues)
We substitute definition of $r_{k+1}$ : MATH MATH MATH MATH We collect the description of the recursion: MATH We eliminate one matrix multiplication by multiplying the last equation by $-A$ and adding $b$ : MATH The recursion starts from MATH and the results are accumulated MATH The equation $\left( \#\right) $ would accumulate numerical errors. Hence, periodically one has to calculate correct residual by using MATH


(Steepest descent) Start from any $x_{0}$ . Set MATH

Do the following MATH Periodically replace MATH before the step MATH .

Notation. Index. Contents.

Copyright 2007