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Solving simultaneous algebra equations
Thank you for visiting our site! You landed on this page because you entered a search term similar to this:
simultaneous equation solver quadratic 3 unknowns, here's the result:
Usually there will be no set of
x_{i} which exactly satisfies (1).
Let us define an error vector e_{j} by
| |
(2) |
It simplifies the development to rewrite this equation as follows
(a trick I learned from John P. Berg).
| |
(3) |
We may abbreviate this equation as
| |
(4) |
where
B is the matrix containing
c and
a.
The
ith error may be written
as a dot product and either vector may be written as the column
Now we will minimize the sum squared error
E defined as
| |
(5) |
The summation may be brought inside the constants
| |
(6) |
The matrix in the center,
call it
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) |
Setting to zero the derivative with respect to
x_{1}, we get
| |
(8) |
Since
r_{ij} =
r_{ji},
both terms on the right are equal.
Thus (8) may be written
| |
(9) |
Likewise,
differentiating with respect to
x_{2} gives
| |
(10) |
Equations (9) and (10) may be combined
| |
(11) |
This form is two equations in two unknowns.
One might write it in the more conventional form
| |
(12) |
The matrix of (11) lacks
only a top row to be equal to the matrix
of (7).
To give it that row,
we may augment (11) by
| |
(13) |
where (13) may be regarded
as a definition of a new variable
v.
Putting (13) on top of (11) we get
| |
(14) |
The solution
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) |
The final equation (14) splits into
| |
(17) |
| (18) |
where (18) represents
simultaneous equations to be solved for
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) |
Multiplying (20) by
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) |
where
^{*} denotes complex-conjugate transpose.
Now (22) gives
| |
(24) |
which is just the complex form of (18).
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) |
When
i has greater range than 3
we have more equations than unknowns.
In this example, (14) takes the form
| |
(26) |
This may be solved by standard methods for
a,
b, and
c.
The last three rows of (26) may be written
| |
(27) |