Returning to the example of substituting the pattern ‘sin(x)^2 + cos(x)^2’ with 1, we saw that the rule ‘opt(a) sin(x)^2 + opt(a) cos(x)^2 := a’ does a good job of finding suitable cases. Another solution would be to use the rule ‘cos(x)^2 := 1 - sin(x)^2’, followed by algebraic simplification if necessary. This rule will be the most effective way to do the job, but at the expense of making some changes that you might not desire.
Another algebraic rewrite rule is ‘exp(x+y) := exp(x) exp(y)’. To make this work with the j r command so that it can be easily targeted to a particular exponential in a large formula, you might wish to write the rule as ‘select(exp(x+y)) := select(exp(x) exp(y))’. The ‘select’ markers will be ignored by the regular a r command (see Selections with Rewrite Rules).
A surprisingly useful rewrite rule is ‘a/(b-c) := a*(b+c)/(b^2-c^2)’. This will simplify the formula whenever ‘b’ and/or ‘c’ can be made simpler by squaring. For example, applying this rule to ‘2 / (sqrt(2) + 3)’ yields ‘6:7 - 2:7 sqrt(2)’ (assuming Symbolic mode has been enabled to keep the square root from being evaluated to a floating-point approximation). This rule is also useful when working with symbolic complex numbers, e.g., ‘(a + b i) / (c + d i)’.
As another example, we could define our own “triangular numbers” function
with the rules ‘[tri(0) := 0, tri(n) := n + tri(n-1) :: n>0]’. Enter
this vector and store it in a variable: s t trirules. Now, given
a suitable formula like ‘tri(5)’ on the stack, type ‘a r trirules’
to apply these rules repeatedly. After six applications, a r will
stop with 15 on the stack. Once these rules are debugged, it would probably
be most useful to add them to EvalRules so that Calc will evaluate
the new tri function automatically. We could then use Z K on
the keyboard macro ' tri($) RET to make a command that applies
tri to the value on the top of the stack. See Programming.
The following rule set, contributed by François
Pinard, implements quaternions, a generalization of the concept of
complex numbers. Quaternions have four components, and are here
represented by function calls ‘quat(w, [x, y,
z])’ with “real part” w and the three “imaginary” parts
collected into a vector. Various arithmetical operations on quaternions
are supported. To use these rules, either add them to EvalRules,
or create a command based on a r for simplifying quaternion
formulas. A convenient way to enter quaternions would be a command
defined by a keyboard macro containing: ' quat($$$$, [$$$, $$, $])
RET.
[ quat(w, x, y, z) := quat(w, [x, y, z]),
quat(w, [0, 0, 0]) := w,
abs(quat(w, v)) := hypot(w, v),
-quat(w, v) := quat(-w, -v),
r + quat(w, v) := quat(r + w, v) :: real(r),
r - quat(w, v) := quat(r - w, -v) :: real(r),
quat(w1, v1) + quat(w2, v2) := quat(w1 + w2, v1 + v2),
r * quat(w, v) := quat(r * w, r * v) :: real(r),
plain(quat(w1, v1) * quat(w2, v2))
:= quat(w1 * w2 - v1 * v2, w1 * v2 + w2 * v1 + cross(v1, v2)),
quat(w1, v1) / r := quat(w1 / r, v1 / r) :: real(r),
z / quat(w, v) := z * quatinv(quat(w, v)),
quatinv(quat(w, v)) := quat(w, -v) / (w^2 + v^2),
quatsqr(quat(w, v)) := quat(w^2 - v^2, 2 * w * v),
quat(w, v)^k := quatsqr(quat(w, v)^(k / 2))
:: integer(k) :: k > 0 :: k % 2 = 0,
quat(w, v)^k := quatsqr(quat(w, v)^((k - 1) / 2)) * quat(w, v)
:: integer(k) :: k > 2,
quat(w, v)^-k := quatinv(quat(w, v)^k) :: integer(k) :: k > 0 ]
Quaternions, like matrices, have non-commutative multiplication.
In other words, ‘q1 * q2 = q2 * q1’ is not necessarily true if
‘q1’ and ‘q2’ are quat forms. The ‘quat*quat’
rule above uses plain to prevent Calc from rearranging the
product. It may also be wise to add the line ‘[quat(), matrix]’
to the Decls matrix, to ensure that Calc’s other algebraic
operations will not rearrange a quaternion product. See Declarations.
These rules also accept a four-argument quat form, converting
it to the preferred form in the first rule. If you would rather see
results in the four-argument form, just append the two items
‘phase(2), quat(w, [x, y, z]) := quat(w, x, y, z)’ to the end
of the rule set. (But remember that multi-phase rule sets don’t work
in EvalRules.)