Using | Solve | And |7T| In Complex Mode; Acuracy In Complex Mode - HP -15C Advanced Functions Handbook

SectionS: Calculating in Complex Mode 73
Using | SOLVE | and | / ? | in Complex Mode
The | SOLVE | and [7T| functions use algorithms that sample your
function at values along the real axis. In Complex mode, the
| SOLVE | and [W\s operate with only the real stack, even
though your function subroutine may have complex computations
in it.
For example, | SOLVE | will not search for the roots of a complex
function, but rather will sample the function on the real axis and
search for a zero of the function's real part. Similarly, [7T| computes
the integral of the function's real part along an interval on the real
axis. These operations are useful in various applications, such as
calculating contour integrals and complex potentials. (Refer to
Applications at the end of this section.)
Accuracy in Complex Mode
Because complex numbers have both real components and
imaginary components, the accuracy of complex calculations takes
on another dimension compared to real-valued calculations.
When dealing with real numbers, an approximation Xis close to x
if the relative difference E(X,x) = \(X — x)/x\s small. This relates
directly to the number of correct significant digits of the
approximation X. That is, if E(X,x) < 5 X 10"", then there are at
least n significant digits. For complex numbers, define E(Z,z) =
\(Z — z ) / z \ This does not relate directly to the number of correct
digits in each component of Z, however.
For example, if E(X,x) and E( Y,y) are both small, then for z =
x + iy, E(Z,z) must also be small. That is, if E(X,x) < s and
E( Y,y) < s, then E(Z,z) < s. But consider z = 10
10
+ i and Z = 10
10
.
The imaginary component of Z is far from accurate, and yet
E(Z,z) < 10"
10
. Even though the imaginary components of z and Z
are completely different, in a relative sense z and Z are extremely
close.
There is a simple, geometric interpretation of the complex relative
error. Any approximation Z of z satisfies E(Z,z) < s (where s is a
positive real number) if and only if Z lies inside the circle of radius
s\z\d at 2 in the complex plane.