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.

## Method of steepest descent.

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

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

 (Orthogonality of residues)
We substitute definition of : We collect the description of the recursion: We eliminate one matrix multiplication by multiplying the last equation by and adding : The recursion starts from and the results are accumulated The equation would accumulate numerical errors. Hence, periodically one has to calculate correct residual by using

Algorithm

(Steepest descent) Start from any . Set

Do the following Periodically replace before the step .

 Notation. Index. Contents.