Sparse coding vector

Jacobson, M.Z., and R.P. Turco, SMVGEAR A sparse-matrix vectorized gear code for atmospheric models. Atmos Environ 28A, 273, 1994. 

A common approach for sparse coding (see for example Lewicki and Sejnowski, 2000) is to use mathematical optimisation to find the best code for signal/image representation. Before coming to the optimisation function, we need to define the norms of a vector. 

An obvious disadvantage of the procedure outlined above is that a relatively large amount of memory is needed. It should also be noted that the method is most advantageous for complete reference functions. For simple reference wavefunctions the matrices Aij, C, etc., become very sparse. This sparsity cannot be exploited fully in a vectorized computer code. It may, therefore, be more efficient to use other techniques in such cases. 

The recent application of sparse-matrix techniques combined with computer optimization (vectorization) techniques has, however, improved the speed of Gear s code substantially, so that this advanced algorithm can now be used to study complex problems in multi-dimensional models (see e.g., the SMVGEAR package developed by Jacobson (1995 1998) and Jacobson and Turco, 1994). 

Let us solve the problem Ax = b, where A has been declared as above and A is a vector containing all ones. Then, we obtain x quickly, taking advantage of the sparse, banded structure of, by typing the code... 

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