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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 conjugate gradients.

eeping the directions MATH (see the section ( Method of conjugate directions )) is memory consuming and the procedure for calculation of such vectors is expensive. According to the formula ( Orthogonality of residues 2 ) the vectors MATH are linearly independent. We take MATH to be initial point of the Gram-Schmidt orthogonalization leading to MATH . According to the section ( Gram-Schmidt orthogonalization ), MATH and according to the summary ( Conjugate directions ) MATH Thus MATH MATH We continue

MATH (Conjugate gradient residue selection)
MATH According to the formula ( Orthogonality of residues 2 ) MATH We conclude MATH MATH Therefore, when we conduct $k$ -th step of Gram-Schmidt MATH -orthogonalization: MATH only one term is non-zero in the sum: MATH We would like to remove matrix multiplications from the above relationship. We set $k:=k-1$ in the equation $\left( \#\right) $ : MATH and apply the operation MATH . Then MATH According to MATH , MATH hence MATH According to the summary ( Conjugate directions ), MATH We combine the last two relationships: MATH thus MATH We substitute it into $\left( \&\right) $ : MATH By MATH -orthogonality of $d_{k-2}$ and $r_{k-1}$ we also have MATH thus MATH We collect the results.


(Conjugate gradients) Start from any $x_{0}$ . Set MATH For $k=0,...,n-1$ do MATH To avoid accumulation of round off errors, occasionally restart with $\left( @\right) $ using last $x_{k}$ as $x_{0}$ . Violation of MATH -orthogonality of MATH is the criteria of error accumulation. Use condition of the type MATH to stop.

Notation. Index. Contents.

Copyright 2007