# Overview of Matrix and Vector Operations

Contents:
  Basic Linear Algebra
  Advanced Functions
  Submatrices, Subvectors
  Speed Improvements

### Definitions:

`A, B, C` | are matrices
`u, v, w` | are vectors
`i, j, k` | are integer values
`t, t1, t2` | are scalar values
`r, r1, r2` | are ranges, e.g. `range(0, 3)`
`s, s1, s2` | are slices, e.g. `slice(0, 1, 3)`

## Basic Linear Algebra

### standard operations: addition, subtraction, multiplication by a scalar

    C = A + B; C = A - B; C = -A;
    w = u + v; w = u - v; w = -u;
    C = t * A; C = A * t; C = A / t;
    w = t * u; w = u * t; w = u / t;

### computed assignements

    C += A; C -= A;
    w += u; w -= u;
    C *= t; C /= t;
    w *= t; w /= t;

### inner, outer and other products

    t = inner_prod(u, v);
    C = outer_prod(u, v);
    w = prod(A, u); w = prod(u, A); w = prec_prod(A, u); w = prec_prod(u, A);
    C = prod(A, B); C = prec_prod(A, B);
    w = element_prod(u, v); w = element_div(u, v);
    C = element_prod(A, B); C = element_div(A, B);

### transformations

    w = conj(u); w = real(u); w = imag(u);
    C = trans(A); C = conj(A); C = herm(A); C = real(A); C = imag(A);

## Advanced functions

### norms

    t = norm_inf(v); i = index_norm_inf(v);
    t = norm_1(v);   t = norm_2(v);
    t = norm_inf(A); i = index_norm_inf(A);
    t = norm_1(A);   t = norm_frobenius(A);

### products

    axpy_prod(A, u, w, true);  // w = A * u
    axpy_prod(A, u, w, false); // w += A * u
    axpy_prod(u, A, w, true);  // w = trans(A) * u
    axpy_prod(u, A, w, false); // w += trans(A) * u
    axpy_prod(A, B, C, true);  // C = A * B
    axpy_prod(A, B, C, false); // C += A * B

Note: The last argument (`bool init`) of `axpy_prod` is optional. Currently it defaults to `true`, but this may change in the future. Set the `init` to `true` is equivalent to call `w.clear()` before `axpy_prod`. Up to now there are some specialisation for compressed matrices that give a large speed up compared to `prod`.

    w = block_prod<matrix_type, 64> (A, u); // w = A * u
    w = block_prod<matrix_type, 64> (u, A); // w = trans(A) * u
    C = block_prod<matrix_type, 64> (A, B); // w = A * B

Note: The blocksize can be any integer. However, the total speed depends very strong on the combination of blocksize, CPU and compiler. The function `block_prod` is designed for large dense matrices.

### rank-k updates

    opb_prod(A, B, C, true);  // C = A * B
    opb_prod(A, B, C, false); // C += A * B

Note: The last argument (`bool init`) of `opb_prod` is optional. Currently it defaults to `true`, but this may change in the future. This function may give a speedup if `A` has less columns than rows, because the product is computed as a sum of outer products.

## Submatrices, Subvectors

Note: A range `r = range(start, stop)` contains all indices `i` with `start <= i < stop`. A slice is something more general. The slice `s = slice(start, stride, size)` contains the indices `start, start+stride, ..., start+(size-1)*stride`. The stride can be 0 or negative! If `start >= stop` for a range or `size == 0` for a slice then it contains no elements.

    w = project(u, r);         // a subvector of u specifed by the index range r
    w = project(u, s);         // a subvector of u specifed by the index slice s
    C = project(A, r1, r2);    // a submatrix of A specified by the two index ranges r1 and r2
    C = project(A, s1, s2);    // a submatrix of A specified by the two index slices s1 and s2
    w = row(A, i); w = column(A, j);    // a row or column of matrix as a vector

There are to more ways to access some matrix elements as a vector:

    matrix_vector_range<matrix_type> (A, r1, r2);
    matrix_vector_slice<matrix_type> (A, s1, s2);

Note: These matrix proxies take a sequence of elements of a matrix and allow you to access these as a vector. In particular `matrix_vector_slice` can do this in a very general way. `matrix_vector_range` is less useful as the elements must lie along a diagonal.

Example: To access the first two elements of a sub column of a matrix we access the row with a slice with stride 1 and the column with a slice with stride 0 thus:
`matrix_vector_slice<matrix_type> (A, slice(0,1,2), slice(0,0,2));`

## Speed improvements

### Matrix / Vector assignment

If you know for sure that the left hand expression and the right hand expression have no common storage, then assignment has no aliasing. A more efficient assignment can be specified in this case:

    noalias(C) = prod(A, B);

This avoids the creation of a temporary matrix that is required in a normal assignment. 'noalias' assignment requires that the left and right hand side be size conformant.

### Sparse element access

The matrix element access function `A(i1,i2)` or the equivalent vector element access functions (`v(i) or v[i]`) usually create 'sparse element proxies' when applied to a sparse matrix or vector. These proxies allow access to elements without having to worry about nasty C++ issues where references are invalidated.

These 'sparse element proxies' can be implemented more efficiently when applied to `const` objects. Sadly in C++ there is no way to distinguish between an element access on the left and right hand side of an assignment. Most often elements on the right hand side will not be changed and therefore it would be better to use the `const` proxies. We can do this by making the matrix or vector `const` before accessing it's elements. For example:

    value = const_cast<const VEC&>(v)[i];   // VEC is the type of V

If more then one element needs to be accessed `const_iterator`'s should be used in preference to `iterator`'s for the same reason. For the more daring 'sparse element proxies' can be completely turned off in uBLAS by defining the configuration macro `BOOST_UBLAS_NO_ELEMENT_PROXIES`.

Copyright (©) 2000-2004 Joerg Walter, Mathias Koch, Gunter Winkler, Michael Stevens
Permission to copy, use, modify, sell and distribute this document is granted provided this copyright notice appears in all copies. This document is provided ``as is'' without express or implied warranty, and with no claim as to its suitability for any purpose.

Last revised: 2004-08-09