When there are more linear equations than unknowns,
it is usually impossible
to find a solution which satisfies all the equations.
Then one often looks for a solution
which approximately satisfies all the equations.
Let **a** and **c** be known
and **x** be unknown in the following set of equations
where there are more equations than unknowns.

(1) |

Usually there will be no set of
*x*_{i} which exactly satisfies (1).
Let us define an error vector *e*_{j} by

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

Comparing (15) with (7),
we see that *v* is the minimum value of *E*.

Occasionally, it is more convenient to have the essential equations in partitioned matrix form. In partitioned matrix form, we have for the error (6)

(16) |

(17) | ||

(18) |

In physical science applications,
the variable *z*_{j} is frequently a complex
variable, say *z*_{j} = *x*_{j} + *iy*_{j}.
It is always possible to go through the
foregoing analyses,
treating the problem as though *x*_{i} and *y*_{i} were
real independent variables.
There is a considerable gain in simplicity and a
saving in computational effort
by treating *z*_{j} as a single complex variable.
The error *E* may be regarded
as a function of either *x*_{j} and *y*_{j} or
*z*_{j} and .
In general but we will
treat the case *N* = 1 here
and leave the general case for the Exercises.
The minimum is found where

(19) | ||

(20) |

(21) | ||

(22) |

However,
the usual case is that *E* is a positive real quadratic function of
*z* and and that
is merely the complex
conjugate of .
Then the two conditions
(21) and (22)
may be replaced by either one of them.
Usually,
when working with complex variables we are minimizing a positive
quadratic form like

(23) |

(24) |

Let us consider an example.
Suppose a set of wave arrival times *t*_{i} is
measured at sensors located on the *x* axis at points *x*_{i}.
Suppose the wavefront is to be fitted to a
parabola .Here,
the *x*_{i} are knowns and *a*, *b*, and *c* are unknowns.
For each sensor *i* we have an equation

(25) |

(26) |

The last three rows of (26) may be written

(27) |

- Extend (24) by fitting waves observed in the
*x*,*y*plane to a two-dimensional quadratic. - Let
*y*(*t*) constitute a complex-valued function at successive integer values of*t*. Fit*y*(*t*) to a least-squares straight line where and . Do it two ways: (a) Assume , ,, and are four independent variables, and (b) Assume , , , and are independent variables. (Leave answer in terms of .) - Equation (14) has assumed all quantities are real. Generalize equation (14) to all complex quantities. Verify that the matrix is Hermitian.
- At the
*j*th seismic observatory (latitude*x*_{j}, longitude*y*_{j}) earthquake waves are observed to arrive at time*t*_{j}. It has been conjectured that the earthquake has an origin time*t*, latitude*x*, and longitude*y*. The theoretical travel time may be looked up in a travel time table where*T*is the travel time and is the great circle angle. One has The time residual at the*j*th station, supposing that the earthquake occurred at (*x*,*y*,*t*), is The time residual, supposing that the earthquake occurred at (*x*+*dx*,*y*+*dy*,*t*+*dt*), is Find equations to solve for*dx*,*dy*, and*dt*which minimize the sum-squared time residuals. - Gravity
*g*_{j}has been measured at*N*irregularly spaced points on the surface of the earth (colatitude*x*_{j}, longitude*y*_{j},*j*= 1,*N*). Show that the matrix of the normal equation which fits the data to spherical harmonics may be written as a sum of a column times its transpose, as in the preceding problem. How would the matrix simplify if there were infinitely many uniformly spaced data points? (NOTE: Spherical harmonics*S*are the class of functions for and where*P*^{m}_{n}is an associated Legendre polynomial of degree*n*and order*m*.) - Ocean tides fit sinusoidal functions of
known frequencies quite accurately.
Associated with the tide is an earth tilt.
A complex time series
may be made from the north-south tilt plus
times the east-west tilt.
The observed complex time series may be fitted to an analytical form
.
Find a set of equations which may be
solved for the
*A*_{j}which gives the best fit of the formula to the data. Show that some elements of the normal equation matrix are sums which may be summed analytically. - The general solution to Laplace's equation
in cylindrical coordinates
for a potential field
*P*which vanishes at is given by Find the potential field surrounding a square object at the origin which is at unit potential. Do this by finding*N*of the coefficients*A*_{m}by minimizing the squared difference between and unity integrated around the square. Give the answer in terms of an inverse matrix of integrals. Which coefficients*A*_{m}vanish exactly by symmetry?

10/30/1997