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**simultaneous equation solver quadratic 3 unknowns**, here's the result:

(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) |

**B**is the matrix containing

**c**and

**a**. The

*i*th error may be written as a dot product and either vector may be written as the column

*E*defined as

(5) |

(6) |

*r*

_{ij}, is symmetrical. It is a positive (more strictly, nonnegative) definite matrix because you will never be able to find an

**x**for which

*E*is negative, since

*E*is a sum of squared

*e*

_{i}. We find the

**x**with minimum

*E*by requiring Notice that this will give us exactly one equation for each unknown. In order to clarify the presentation we will specialize (6) to two unknowns.

(7) |

*x*, we get

_{1}(8) |

*r*

_{ij}=

*r*

_{ji}, both terms on the right are equal. Thus (8) may be written

(9) |

*x*gives

_{2}(10) |

(11) |

(12) |

(13) |

*v*. Putting (13) on top of (11) we get

(14) |

**x**of (12) or (14) is that set of

*x*

_{k}for which

*E*is a minimum. To get an interpretation of

*v*, we may multiply both sides by , getting

(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) |

**x**. Equation (18) is what you have to set up in a computer. It is easily remembered by a quick and dirty (very dirty) derivation. That is, we began with the overdetermined equations ;premultiplying by gives which is (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) |

*i*and adding and subtracting these equations, we may express the minimum condition more simply as

(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) |

^{*}denotes complex-conjugate transpose. Now (22) gives

(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) |

*i*has greater range than 3 we have more equations than unknowns. In this example, (14) takes the form

(26) |

*a*,

*b*, and

*c*.

The last three rows of (26) may be written

(27) |