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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.
 B. Method of steepest descent.
 C. Method of conjugate directions.
 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.

eeping the directions (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 are linearly independent. We take to be initial point of the Gram-Schmidt orthogonalization leading to . According to the section ( Gram-Schmidt orthogonalization ), and according to the summary ( Conjugate directions ) Thus We continue

According to the formula ( Orthogonality of residues 2 ) We conclude Therefore, when we conduct -th step of Gram-Schmidt -orthogonalization: only one term is non-zero in the sum: We would like to remove matrix multiplications from the above relationship. We set in the equation : and apply the operation . Then According to , hence According to the summary ( Conjugate directions ), We combine the last two relationships: thus We substitute it into : By -orthogonality of and we also have thus We collect the results.

Algorithm

(Conjugate gradients) Start from any . Set For do To avoid accumulation of round off errors, occasionally restart with using last as . Violation of -orthogonality of is the criteria of error accumulation. Use condition of the type to stop.

 Notation. Index. Contents.