Let us minimize the sum of the squares of the components of the residual vector given by

(39) | ||

(40) |

A contour plot is based on an altitude function of space.
The altitude is the dot product .By finding the lowest altitude,
we are driving the residual vector as close as we can to zero.
If the residual vector reaches zero, then we have solved
the simultaneous equations .In a two-dimensional world the vector has two components,
(*x _{1}* ,

Let us see how a random search-direction can be used to reduce the residual .Let be an abstract vector with the same number of components as the solution ,and let contain arbitrary or random numbers. We add an unknown quantity of vector to the vector ,and thereby create :

(41) |

(42) | ||

(43) | ||

(44) |

Next we adjust to minimize the dot product:

(45) |

(46) |

(47) |

A ``computation template'' for the method of random directions is

A nice thing about the method of random directions is that you do not need to know the adjoint operator .iterate { }

In practice, random directions are rarely used. It is more common to use the gradient direction than a random direction. Notice that a vector of the size of is

(48) |

(49) |

Starting from a model (which may be zero), below is a template of pseudocode for minimizing the residual by the steepest-descent method:

iterate { }

4/27/2004