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

| |
(30) |

| |
(31) |

Fourier-transformed variables are often capitalized.
Here we capitalize vectors transformed by the matrix.
A matrix such as is denoted by **boldface** print.
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 *R* as close as we can to zero.
If the residual vector *R* reaches zero, then we have solved
the simultaneous equations .In a two-dimensional world the vector *x* has two components,
(*x*_{1} , *x*_{2} ).
A contour is a curve of constant in (*x*_{1} , *x*_{2} )-space.
These contours have a statistical interpretation as contours
of uncertainty in (*x*_{1} , *x*_{2} ), given measurement errors in *Y*.

Starting from , let us see how a random search
direction can be used to try to reduce the residual.
Let *g* be an abstract vector with the same number of components as
the solution *x*,
and let *g* contain arbitrary or random numbers.
Let us add an unknown quantity of vector *g* to vector *x*,
thereby changing *x* to .The new residual *R*+*dR* becomes

| |
(32) |

| (33) |

| (34) |

We seek to minimize the dot product
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(35) |

Setting to zero the derivative with respect to gives
| |
(36) |

Geometrically and algebraically
the new residual is
perpendicular to the ``fitting function'' *G*.
(We confirm this by substitution leading to
In practice, random directions are rarely used.
It is more common to use the **gradient** vector.
Notice also that a vector of the size of *x* is

| |
(37) |

Notice also that this vector can be found by taking the gradient
of the size of the residuals:
| |
(38) |

Descending by use of the gradient vector is called
``the method of **steepest descent**."

** Next:** Conditioning the gradient
** Up:** ITERATIVE METHODS
** Previous:** ITERATIVE METHODS
Stanford Exploration Project

10/21/1998