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8 Some Fundamenurl Numerical Methods
degrees, then magnitude. For the operation Z, + Z" it is necessary to enter
degZ" magZ" degZ" magZ, and press key B to see mag(Z,+Z,) in the X
register (and the angle in the Y register by pressing key A to swap X and Y
registers). Complex subtraction depends on a feature of the HP calculator in
which a negative-magnitude number adds 180 degrees to the angle during the
next operation. Thus a separate key for complex subtraction is not required;
just key in the sequence for Z, +Z" but press the CHS (change sign) key
before pressing B (+) key. The answer is Z,-Z2' A complex-division key is
made unnecessary by providing the complex inverse function I/Z on key C.
Thus to compute Z,/Z" the stack entries (in order) are: degZ" magZ"
degZ" magZ,. Then press key C to obtain I/Z, (the answer is placed
properly in stack registers X and Y without disturbing registers Z and T),
followed by pressing key D for tbe complex multiplication. Again, the answer
appears in stack positions X and Y. Example 2.1 shows that manual or
programmed steps with complex numbers are as easy as with real numbers.
Example 2./.
Consider the bilinear function from Chapter Seven:
a,Z+a,
W= a,Z+1 . (2.1)
All variables may be complex; suppose that a, =0.6 /75°, a,=0.18 / -23°,
and a,=1.4 /130°. Given Z=0.5 /60°, what is w? The manual or programmed
steps are the same: enter Z in the stack and also store its angle and
magnitude in two spare registers. Then enter a, and multiply, enter 0 degrees
and unity magnitude and add, saving the two parts of the denominator value
in two more spare registers. The numerator is computed in the same way, the
denominator value is recalled into the stack and inverted, and the two
complex numbers in the stack are multiplied. The correct answer is w=
0.4473 /129.5°. Normally, a given set of coefficients (a" a" and a,) are fixed,
and a sequence of Z values are input into the program. A helpful hint for
evaluating bilinear functions is to rewrite them by doing long division on
(2.1):
(2.2)
Then store a,/a" a 2 , and a,. Now the operations for evaluating (2.2) do not
require storing Z, although a zero denominator value should be anticipated by
always adding I.E - 9 (0.000000001) to its magnitude before inverting. If there
is a fourth complex coefficient in place of unity in the denominator of (2.1),
the standard form of (2.1) should be obtained by first dividing the other
coefficients by the fourth coefficient.
2.2. Linear Systems of Equations
Every engineering discipline requires the solution of sets of linear equations
with real coefficients; this will also be required in Section 2.5 of this chapter.