Usenet 1994 October/usenetsourcesnewsgroupsinfomagicoctober1994disk2.iso/misc/volume24/gnucalc/part46

Text File | 1991-11-01 | 55.5 KB | 1,209 lines

Newsgroups: comp.sources.misc
From: [e-mail] (David Gillespie)
Subject:  v24i094:  gnucalc - GNU Emacs Calculator, v2.00, Part46/56
Message-ID: <[e-mail]>
X-Md4-Signature: 978bcd10a1385fa395f7fac5ddec7ca2
Date: Fri, 1 Nov 1991 18:38:40 GMT
Approved: [e-mail]

Submitted-by: [e-mail] (David Gillespie)
Posting-number: Volume 24, Issue 94
Archive-name: gnucalc/part46
Environment: Emacs
Supersedes: gmcalc: Volume 13, Issue 27-45

---- Cut Here and unpack ----
#!/bin/sh
# do not concatenate these parts, unpack them in order with /bin/sh
# file calc.texinfo continued
#
if test ! -r _shar_seq_.tmp; then
    echo 'Please unpack part 1 first!'
    exit 1
fi
(read Scheck
 if test "$Scheck" != 46; then
    echo Please unpack part "$Scheck" next!
    exit 1
 else
    exit 0
 fi
) < _shar_seq_.tmp || exit 1
if test ! -f _shar_wnt_.tmp; then
    echo 'x - still skipping calc.texinfo'
else
echo 'x - continuing file calc.texinfo'
sed 's/^X//' << 'SHAR_EOF' >> 'calc.texinfo' &&
the function names @code{thecoefs} and @code{thefactors} are used only
as placeholders; there are no actual Calc functions by those names.)
X
@kindex H a f
@tindex factors
The @kbd{H a f} [@code{factors}] command also factors a polynomial,
but it returns a list of factors instead of an expression which is the
product of the factors.  Each factor is represented by a sub-vector
of the factor, and the power with which it appears.  For example,
@cite{x^5 + x^4 - 33 x^3 + 63 x^2} factors to @cite{(x + 7) x^2 (x - 3)^2}
in @kbd{a f}, or to @cite{[ [x, 2], [x+7, 1], [x-3, 2] ]} in @kbd{H a f}.
If there is an overall numeric factor, it always comes first in the list.
The functions @code{factor} and @code{factors} allow a second argument
when written in algebraic form; @samp{factor(x,v)} factors @cite{x} with
respect to the specific variable @cite{v}.  The default is to factor with
respect to all the variables that appear in @cite{x}.
X
@kindex a c
@pindex calc-collect
@tindex collect
The @kbd{a c} (@code{calc-collect}) [@code{collect}] command rearranges a
formula as a
polynomial in a given variable, ordered in decreasing powers of that
variable.  For example, given @cite{1 + 2 x + 3 y + 4 x y^2} on
the stack, @kbd{a c x} would produce @cite{(2 + 4 y^2) x + (1 + 3 y)},
and @kbd{a c y} would produce @cite{(4 x) y^2 + 3 y + (1 + 2 x)}.
The polynomial will be expanded out using the distributive law as
necessary:  Collecting @cite{x} in @cite{(x - 1)^3} produces
@cite{x^3 - 3 x^2 + 3 x - 1}.  Terms not involving @cite{x} will
not be expanded.
X
The ``variable'' you specify at the prompt can actually be any
expression: @kbd{a c ln(x+1)} will collect together all terms multiplied
by @samp{ln(x+1)} or integer powers thereof.  If @samp{x} also appears
in the formula in a context other than @samp{ln(x+1)}, @kbd{a c} will
treat those occurrences as unrelated to @samp{ln(x+1)}, i.e., as constants.
X
@kindex a x
@pindex calc-expand
@tindex expand
The @kbd{a x} (@code{calc-expand}) [@code{expand}] command expands an
expression by applying the distributive law everywhere.  It applies to
products, quotients, and powers involving sums.  By default, it fully
distributes all parts of the expression.  With a numeric prefix argument,
the distributive law is applied only the specified number of times, then
the partially expanded expression is left on the stack.
X
The @kbd{a x} and @kbd{j D} commands are somewhat redundant.  Use
@kbd{a x} if you want to expand all products of sums in your formula.
Use @kbd{j D} if you want to expand a particular specified term of
the formula.  There is an exactly analogous correspondence between
@kbd{a f} and @kbd{j M}.  (The @kbd{j D} and @kbd{j M} commands
also know many other kinds of expansions, such as
@samp{exp(a + b) = exp(a) exp(b)}.)
X
Calc's automatic simplifications will sometimes reverse a partial
expansion.  For example, the first step in expanding @cite{(x+1)^3} is
to write @cite{(x+1) (x+1)^2}.  If @kbd{a x} stops there and tries
to put this formula onto the stack, though, Calc will automatically
simplify it back to @cite{(x+1)^3} form.  The solution is to turn
simplification off first (@pxref{Simplification Modes}), or to run
@kbd{a x} without a numeric prefix argument so that it expands all
the way in one step.
X
@kindex a a
@pindex calc-apart
@tindex apart
The @kbd{a a} (@code{calc-apart}) [@code{apart}] command expands a
rational function by partial fractions.  A rational function is the
quotient of two polynomials; @code{apart} pulls this apart into a
sum of rational functions with simple denominators.  In algebraic
notation, the @code{apart} function allows a second argument that
specifies which variable to use as the ``base''; by default, Calc
chooses the base variable automatically.
X
@kindex a n
@pindex calc-normalize-rat
@tindex nrat
The @kbd{a n} (@code{calc-normalize-rat}) [@code{nrat}] command
attempts to arrange a formula into a quotient of two polynomials.
For example, given @cite{1 + (a + b/c) / d}, the result would be
@cite{(b + a c + c d) / c d}.  The quotient is reduced, so that
@kbd{a n} will simplify @cite{(x^2 + 2x + 1) / (x^2 - 1)} by dividing
out the common factor @cite{x + 1}, yielding @cite{(x + 1) / (x - 1)}.
X
@kindex a \
@pindex calc-poly-div
@tindex pdiv
The @kbd{a \} (@code{calc-poly-div}) [@code{pdiv}] command divides
two polynomials @cite{u} and @cite{v}, yielding a new polynomial
@cite{q}.  If several variables occur in the inputs, the inputs are
considered multivariate polynomials.  (Calc divides by the variable
with the largest power in @cite{u} first, or, in the case of equal
powers, chooses the variables in alphabetical order.)  For example,
dividing @cite{x^2 + 3 x + 2} by @cite{x + 2} yields @cite{x + 1}.
The remainder from the division, if any, is reported at the bottom
of the screen and is also placed in the Trail along with the quotient.
X
Using @code{pdiv} in algebraic notation, you can specify the particular
variable to be used as the base:  `@t{pdiv(}@i{a}@t{,}@i{b}@t{,}@i{x}@t{)}'.
If @code{pdiv} is given only two arguments (as is always the case with
the @kbd{a \} command), then it does a multivariate division as outlined
above.
X
@kindex a %
@pindex calc-poly-rem
@tindex prem
The @kbd{a %} (@code{calc-poly-rem}) [@code{prem}] command divides
two polynomials and keeps the remainder @cite{r}.  The quotient
@cite{q} is discarded.  For any formulas @cite{a} and @cite{b}, the
results of @kbd{a \} and @kbd{a %} satisfy @cite{a = q b + r}.
(This is analogous to plain @kbd{\} and @kbd{%}, which compute the
integer quotient and remainder from dividing two numbers.)
X
@kindex a /
@kindex H a /
@pindex calc-poly-div-rem
@tindex pdivrem
@tindex pdivide
The @kbd{a /} (@code{calc-poly-div-rem}) [@code{pdivrem}] command
divides two polynomials and reports both the quotient and the
remainder as a vector @cite{[q, r]}.  The @kbd{H a /} [@code{pdivide}]
command divides two polynomials and constructs the formula
@cite{q + r/b} on the stack.  (Naturally if the remainder is zero,
this will immediately simplify to @cite{q}.)
X
@kindex a g
@pindex calc-poly-gcd
@tindex pgcd
The @kbd{a g} (@code{calc-poly-gcd}) [@code{pgcd}] command computes
the greatest common divisor of two polynomials.  (The GCD actually
is unique only to within a constant multiplier; Calc attempts to
choose a GCD which will be unsurprising.)  For example, the @kbd{a n}
command uses @kbd{a g} to take the GCD of the numerator and denominator
of a quotient, then divides each by the result using @kbd{a \}.  (The
definition of GCD ensures that this division can take place without
leaving a remainder.)
X
While the polynomials used in operations like @kbd{a /} and @kbd{a g}
often have integer coefficients, this is not required.  Calc can also
deal with polynomials over the rationals or floating-point reals.
Polynomials with modulo-form coefficients are also useful in many
applications; if you enter @samp{(x^2 + 3 x - 1) mod 5}, Calc
automatically transforms this into a polynomial over the field of
integers mod 5:  @samp{(1 mod 5) x^2 + (3 mod 5) x + (4 mod 5)}.
X
Congratulations and thanks go to Ove Ewerlid
(@code{ewerlid@@mizar.DoCS.UU.SE}), who contributed many of the
polynomial routines used in the above commands.
X
@xref{Decomposing Polynomials}, for several useful functions for
extracting the individual coefficients of a polynomial.
X
@node Calculus, Solving Equations, Polynomials, Algebra
@section Calculus
X
@noindent
The following calculus commands do not automatically simplify their
inputs or outputs using @code{calc-simplify}.  You may find it helps
to do this by hand by typing @kbd{a s} or @kbd{a e}.  It may also help
to use @kbd{a x} and/or @kbd{a c} to arrange a result in the most
readable way.
X
@menu
* Differentiation::
* Integration::
* Customizing the Integrator::
* Numerical Integration::
* Taylor Series::
@end menu
X
@node Differentiation, Integration, Calculus, Calculus
@subsection Differentiation
X
@kindex a d
@kindex H a d
@pindex calc-derivative
@tindex deriv
@tindex tderiv
@noindent
The @kbd{a d} (@code{calc-derivative}) [@code{deriv}] command computes
the derivative of the expression on the top of the stack with respect to
some variable, which it will prompt you to enter.  Normally, variables
in the formula other than the specified differentiation variable are
considered constant, i.e., @samp{deriv(y,x)} is reduced to zero.  With
the Hyperbolic flag, the @code{tderiv} (total derivative) operation is used
instead, in which derivatives of variables are not reduced to zero
unless those variables are known to be ``constant,'' i.e., independent
of any other variables.  (The built-in special variables like @code{pi}
are considered constant, as are variables that have been declared
@code{const}; @pxref{Declarations}.)
X
With a numeric prefix argument @var{n}, this computes the @var{n}th
derivative.
X
When working with trigonometric functions, it is best to switch to
radians mode first (with @kbd{m r}).  The derivative of @samp{sin(x)}
in degrees is @samp{(pi/180) cos(x)}, probably not the expected
answer!
X
If you use the @code{deriv} function directly in an algebraic formula,
you can write @samp{deriv(f,x,x0)} which represents the derivative
of @cite{f} with respect to @cite{x}, evaluated at the point @cite{x=x0}.
X
If the formula being differentiated contains functions which Calc does
not know, the derivatives of those functions are produced by adding
primes (apostrophe characters).  For example, @samp{deriv(f(2x), x)}
produces @samp{2 f'(2 x)}, where the function @code{f'} represents the
derivative of @code{f}.
X
For functions you have defined with the @kbd{Z F} command, Calc expands
the functions according to their defining formulas unless you have
also defined @code{f'} suitably.  For example, suppose we define
@samp{sinc(x) = sin(x)/x} using @kbd{Z F}.  If we then differentiate
the formula @samp{sinc(2 x)}, the formula will be expanded to
@samp{sin(2 x) / (2 x)} and differentiated.  However, if we also
define @samp{sinc'(x) = dsinc(x)}, say, then Calc will write the
result as @samp{2 dsinc(x)}.  @xref{Algebraic Definitions}.
X
For multi-argument functions @samp{f(x,y,z)}, the derivative with respect
to the first argument is written @samp{f'(x,y,z)}; derivatives with
respect to the other arguments are @samp{f'2(x,y,z)} and @samp{f'3(x,y,z)}.
Various higher-order derivatives can be formed in the obvious way, e.g.,
@samp{f'@var{}'(x)} (the second derivative of @code{f}) or
@samp{f'@var{}'2'3(x,y,z)} (@code{f} differentiated with respect to each
argument once).@refill
X
@node Integration, Customizing the Integrator, Differentiation, Calculus
@subsection Integration
X
@kindex a i
@pindex calc-integral
@tindex integ
@noindent
The @kbd{a i} (@code{calc-integral}) [@code{integ}] command computes the
indefinite integral of the expression on the top of the stack with
respect to a variable.  The integrator is not guaranteed to work for
all integrable functions, but it is able to integrate several large
classes of formulas.  In particular, any polynomial or rational function
(a polynomial divided by a polynomial) is acceptable.  (Rational functions
don't have to be in explicit quotient form, however; @c{$x/(1+x^{-2}$}
@cite{x/(1+x^-2)}
is not strictly a quotient of polynomials, but it is equivalent to
@cite{x^3/(x^2+1)}, which is.)  Also, square roots of terms involving
@cite{x} and @cite{x^2} may appear in rational functions being
integrated.  Finally, rational functions involving trigonometric or
hyperbolic functions can be integrated.
X
@ifinfo
If you use the @code{integ} function directly in an algebraic formula,
you can also write @samp{integ(f,x,v)} which expresses the resulting
indefinite integral in terms of variable @code{v} instead of @code{x}.
With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
integral from @code{a} to @code{b}.
@end ifinfo
@tex
If you use the @code{integ} function directly in an algebraic formula,
you can also write @samp{integ(f,x,v)} which expresses the resulting
indefinite integral in terms of variable @code{v} instead of @code{x}.
With four arguments, @samp{integ(f(x),x,a,b)} represents a definite
integral $\int_a^b f(x) \, dx$.
@end tex
X
Please note that the current implementation of Calc's integrator sometimes
produces results that are significantly more complex than they need to
be.  For example, the integral Calc finds for @c{$1/(x+\sqrt{x^2+1})$}
@cite{1/(x+sqrt(x^2+1))}
is several times more complicated than the answer Mathematica
returns for the same input, although the two forms are numerically
equivalent.  Also, any indefinite integral should be considered to have
an arbitrary constant of integration added to it, although Calc does not
write an explicit constant of integration in its result.  For example,
Calc's solution for @c{$1/(1+\tan x)$}
@cite{1/(1+tan(x))} differs from the solution given
in the @emph{CRC Math Tables} by a constant factor of @c{$\pi i / 2$}
@cite{pi i / 2},
due to a different choice of constant of integration.
X
The Calculator remembers all the integrals it has done.  If conditions
change in a way that would invalidate the old integrals, they will be
thrown out.  If you suspect this is not happening when it should,
use the @code{calc-flush-caches} command; @pxref{Caches}.
X
@vindex IntegLimit
Calc normally will pursue integration by substitution or integration by
parts up to 3 nested times before abandoning an approach as fruitless.
If the integrator is taking too long, you can lower this limit by storing
a number (like 2) in the variable @code{IntegLimit}.  (The @kbd{s I}
command is a convenient way to edit @code{IntegLimit}.)  If this variable
has no stored value or does not contain a nonnegative integer, a limit
of 3 is used.  The lower this limit is, the greater the chance that Calc
will be unable to integrate a function it could otherwise handle.  Raising
this limit allows the Calculator to solve more integrals, though the time
it takes may grow exponentially.  You can monitor the integrator's actions
by creating an Emacs buffer called @code{*Trace*}.  If such a buffer
exists, the @kbd{a i} command will write a log of its actions there.
X
If you want to manipulate integrals in a purely symbolic way, you can
set the integration nesting limit to 0 to prevent all but fast
table-lookup solutions of integrals.  You might then wish to define
rewrite rules for integration by parts, various kinds of substitutions,
and so on.  @xref{Rewrite Rules}.
X
@node Customizing the Integrator, Numerical Integration, Integration, Calculus
@subsection Customizing the Integrator
X
@vindex IntegRules
@noindent
Calc has two built-in rewrite rules called @code{IntegRules} and
@code{IntegAfterRules} which you can edit to define new integration
methods.  @xref{Rewrite Rules}.  At each step of the integration process,
Calc wraps the current integrand in a call to the fictitious function
@samp{integtry(@var{expr},@var{var})}, where @var{expr} is the
integrand and @var{var} is the integration variable.  If your rules
rewrite this to be a plain formula (not a call to @code{integtry}, then
Calc will use this formula as the integral of @var{expr}.  For example,
the rule @samp{integtry(mysin(x),x) := -mycos(x)} would define a rule to
integrate a function @code{mysin} that acts like the sine function.
Then, putting @samp{4 mysin(2y+1)} on the stack and typing @kbd{a i y}
will produce the integral @samp{-2 mycos(2y+1)}.  Note that Calc has
automatically made various transformations on the integral to allow it
to use your rule; integral tables generally give rules for
@samp{mysin(a x + b)}, but you don't need to use this generality in
your @code{IntegRules}.
X
Your rule may do further integration by calling @code{integ}.  For
example, @samp{integtry(twice(u),x) := twice(integ(u))} allows Calc
to integrate @samp{twice(sin(x))} to get @samp{twice(-cos(x))}.
Note that @code{integ} was called with only one argument.  This notation
is allowed only within @code{IntegRules}; it means ``integrate this
with respect to the same integration variable.''  If Calc is unable
to integrate @code{u}, the integration that invoked @code{IntegRules}
also fails.  Thus integrating @samp{twice(f(x))} fails, returning the
unevaluated integral @samp{integ(twice(f(x)), x)}.  It is still legal
to call @code{integ} with two or more arguments, however; in this case,
if @code{u} is not integrable, @code{twice} itself will still be
integrated:  If the above rule is changed to @samp{... := twice(integ(u,x))},
then integrating @samp{twice(f(x))} will yield @samp{twice(integ(f(x),x))}.
X
If a rule instead produces the formula @samp{integsubst(@var{sexpr},
@var{svar})}, either replacing the top-level @code{integtry} call or
nested anywhere inside the expression, then Calc will apply the
substitution @samp{@var{u} = @var{sexpr}(@var{svar})} to try to
integrate the original @var{expr}.  For example, the rule
@samp{sqrt(a) := integsubst(sqrt(x),x)} says that if Calc ever finds
a square root in the integrand, it should attempt the substitution
@samp{u = sqrt(x)}.  (This particular rule is unnecessary because
Calc always tries ``obvious'' substitutions where @var{sexpr} actually
appears in the integrand.)  The variable @var{svar} may be the same
as the @var{var} that appeared in the call to @code{integtry}, but
it need not be.
X
When integrating according to an @code{integsubst}, Calc uses the
equation solver to find the inverse of @var{sexpr} (if the integrand
refers to @var{var} anywhere except in subexpressions that exactly
match @var{sexpr}).  It uses the differentiator to find the derivative
of @var{sexpr} and/or its inverse (it has two methods that use one
derivative or the other).  You can also specify these items by adding
extra arguments to the @code{integsubst} your rules construct; the
general form is @samp{integsubst(@var{sexpr}, @var{svar}, @var{sinv},
@var{sprime})}, where @var{sinv} is the inverse of @var{sexpr} (still
written as a function of @var{svar}), and @var{sprime} is the
derivative of @var{sexpr} with respect to @var{svar}.  If you don't
specify these things, and Calc is not able to work them out on its
own with the information it knows, then your substitution rule will
work only in very specific, simple cases.
X
Calc applies @code{IntegRules} as if by @kbd{C-u 1 a r IntegRules};
in other words, Calc stops rewriting as soon as any rule in your rule
set succeeds.  (If it weren't for this, the @samp{integsubst(sqrt(x),x)}
example above would keep on adding layers of @code{integsubst} calls
forever!)
X
@vindex IntegSimpRules
Another set of rules, stored in @code{IntegSimpRules}, are applied
every time the integrator uses @kbd{a s} to simplify an intermediate
result.  For example, putting the rule @samp{twice(x) := 2 x} into
@code{IntegSimpRules} would tell Calc to convert the @code{twice}
function into a form it knows whenever integration is attempted.
X
One more way to influence the integrator is to define a function with
the @kbd{Z F} command (@pxref{Algebraic Definitions}).  Calc's
integrator automatically expands such functions according to their
defining formulas, even if you originally asked for the function to
be left unevaluated for symbolic arguments.  (Certain other Calc
systems, such as the differentiator and the equation solver, also
do this.)
X
@vindex IntegAfterRules
Sometimes Calc is able to find a solution to your integral, but it
expresses the result in a way that is unnecessarily complicated.  If
this happens, you can either use @code{integsubst} hints as described
above to try to hint at a more direct path to the desired result, or
you can use @code{IntegAfterRules}.  This is an extra rule set that
runs after the main integrator returns its result; basically, Calc does
an @kbd{a r IntegAfterRules} on the result before showing it to you.
(It also does an @kbd{a s}, without @code{IntegSimpRules}, after that
to further simplify the result.)  For example, Calc's integrator
sometimes produces expressions of the form @samp{ln(1+x) - ln(1-x)};
the default @code{IntegAfterRules} rewrite this into the more readable
form @samp{2 arctanh(x)}.  Note that, unlike @code{IntegRules},
@code{IntegSimpRules} and @code{IntegAfterRules} are applied any number
of times until no further changes are possible.  Rewriting by
@code{IntegAfterRules} occurs only after the main integrator has
finished, not at every step as for @code{IntegRules} and
@code{IntegSimpRules}.
X
@node Numerical Integration, Taylor Series, Customizing the Integrator, Calculus
@subsection Numerical Integration
X
@kindex a I
@pindex calc-num-integral
@tindex ninteg
@noindent
If you want a purely numerical answer to an integration problem, you can
use the @kbd{a I} (@code{calc-num-integral}) [@code{ninteg}] command.  This
command prompts for an integration variable, a lower limit, and an
upper limit.  Except for the integration variable, all other variables
that appear in the integrand formula must have stored values.  (A stored
value, if any, for the integration variable itself is ignored.)
X
Numerical integration works by evaluating your formula at many points in
the specified interval.  Calc uses an ``open Romberg'' method; this means
that it does not evaluate the formula actually at the endpoints (so that
it is safe to integrate @samp{sin(x)/x} from zero, for example).  Also,
the Romberg method works especially well when the function being
integrated is fairly smooth.  If the function is not smooth, Calc will
have to evaluate it at quite a few points before it can accurately
determine the value of the integral.
X
Integration is much faster when the current precision is small.  It is
best to set the precision to the smallest acceptable number of digits
before you use @kbd{a I}.  If Calc appears to be taking too long, press
@kbd{C-g} to halt it and try a lower precision.  If Calc still appears
to need hundreds of evaluations, check to make sure your function is
well-behaved in the specified interval.
X
It is possible for the lower integration limit to be @samp{-inf} (minus
infinity).  Likewise, the upper limit may be plus infinity.  Calc
internally transforms the integral into an equivalent one with finite
limits.
X
@node Taylor Series, , Numerical Integration, Calculus
@subsection Taylor Series
X
@kindex a t
@pindex calc-taylor
@tindex taylor
@noindent
The @kbd{a t} (@code{calc-taylor}) [@code{taylor}] command computes a
power series expansion or Taylor series of a function.  You specify the
variable and the desired number of terms.  You may give an expression of
the form @samp{@var{var} = @var{a}} or @samp{@var{var} - @var{a}} instead
of just a variable to produce a Taylor expansion about the point @var{a}.
You may specify the number of terms with a numeric prefix argument;
otherwise the command will prompt you for the number of terms.  Note that
many series expansions have coefficients of zero for some terms, so you
may appear to get fewer terms than you asked for.@refill
X
If the @kbd{a i} command is unable to find a symbolic integral for a
function, you can get an approximation by integrating the function's
Taylor series.
X
@node Solving Equations, Numerical Solutions, Calculus, Algebra
@section Solving Equations
X
@noindent
@kindex a S
@pindex calc-solve-for
@tindex solve
@cindex Equations, solving
@cindex Solving equations
The @kbd{a S} (@code{calc-solve-for}) [@code{solve}] command rearranges
an equation to solve for a specific variable.  An equation is an
expression of the form @cite{L = R}.  For example, the command @kbd{a S x}
will rearrange @cite{y = 3x + 6} to the form, @cite{x = y/3 - 2}.  If the
input is not an equation, it is treated like an equation of the
form @cite{X = 0}.
X
This command also works for inequalities, as in @cite{y < 3x + 6}.
Some inequalities cannot be solved where the analogous equation could
be; for example, solving @cite{a < b c} for @cite{b} is impossible
without knowing the sign of @cite{c}.  In this case, @kbd{a S} will
produce the result @c{$b \mathbin{\hbox{\code{!=}}} a/c$}
@cite{b != a/c} (using the not-equal-to operator)
to signify that the direction of the inequality was unknown.  The
inequality @c{$a \le b \, c$}
@cite{a <= b c} is not even partially solved.
@xref{Declarations}, for a way to tell Calc that the signs of the
variables in a formula are in fact known.
X
Two useful commands for working with the result of @kbd{a S} are
@kbd{a .} (@pxref{Logical Operations}), which converts @samp{x = y/3 - 2}
to @samp{y/3 - 2}, and @kbd{s l} (@pxref{Let Command}) which evaluates
a formula with @samp{x} set equal to @samp{y/3 - 2}.
X
@menu
* Multiple Solutions::
* Solving Systems of Equations::
* Decomposing Polynomials::
@end menu
X
@node Multiple Solutions, Solving Systems of Equations, Solving Equations, Solving Equations
@subsection Multiple Solutions
X
@noindent
@kindex H a S
@tindex fsolve
Some equations have more than one solution.  The Hyperbolic flag
(@code{H a S}) [@code{fsolve}] tells the solver to report the fully
general family of solutions.  It will invent variables @code{n1},
@code{n2}, @dots{}, which represent independent arbitrary integers, and
@code{s1}, @code{s2}, @dots{}, which represent independent arbitrary
signs (either @i{+1} or @i{-1}).  If you don't use the Hyperbolic
flag, Calc will use zero in place of all arbitrary integers, and plus
one in place of all arbitrary signs.  Note that variables like @code{n1}
and @code{s1} are not given any special interpretation in Calc except by
@code{calc-solve-for} itself.  As usual, you can use the @kbd{s l}
(@code{calc-let}) command to obtain solutions for various actual values
of these variables.
X
For example, @kbd{' x^2 = y @key{RET} H a S x @key{RET}} solves to
get @samp{x = s1 sqrt(y)}, indicating that the two solutions to the
equation are @samp{sqrt(y)} and @samp{-sqrt(y)}.  Another way to
think about it is that the square-root operation is really a
two-valued function; since every Calc function must return a
single result, @code{sqrt} chooses to return the positive result.
Then @kbd{H a S} doctors this result using @code{s1} to indicate
the full set of possible values of the mathematical square-root.
X
There is a similar phenomenon going the other direction:  Suppose
we solve @samp{sqrt(y) = x} for @code{y}.  Calc squares both sides
to get @samp{y = x^2}.  This is correct, except that it introduces
some dubious solutions.  Consider solving @samp{sqrt(y) = -3}:
Calc will report @cite{y = 9} as a valid solution, which is true
in the mathematical sense of square-root, but false (there is no
solution) for the actual Calc positive-valued @code{sqrt}.  This
happens for both @kbd{a S} and @kbd{H a S}.
X
@cindex @code{GenCount} variable
@vindex GenCount
@tindex an
@tindex as
If you store a positive integer in the Calc variable @code{GenCount},
then Calc will generate formulas of the form @samp{as(@var{n})} for
arbitrary signs, and @samp{an(@var{n})} for arbitrary integers,
where @var{n} represents succeeding values taken by incrementing
@code{GenCount} by one.  While the normal arbitrary sign and
integer symbols start over at @code{s1} and @code{n1} with each
new Calc command, the @code{GenCount} approach will give each
arbitrary value a name that is unique throughout the entire Calc
session.  Also, the arbitrary values are function calls instead
of variables, which is advantageous in some cases.  For example,
you can make a rewrite rule that recognizes all arbitrary signs
using a pattern like @samp{as(n)}.  The @kbd{s l} command only works
on variables, but you can use the @kbd{a b} (@code{calc-substitute})
command to substitute actual values for function calls like @samp{as(3)}.
X
The @kbd{s G} (@code{calc-edit-GenCount}) command is a convenient
way to create or edit this variable.  Press @kbd{M-# M-#} to finish.
X
If you have not stored a value in @code{GenCount}, or if the value
in that variable is not a positive integer, the regular
@code{s1}/@code{n1} notation is used.
X
@kindex I a S
@kindex H I a S
@tindex finv
@tindex ffinv
With the Inverse flag, @kbd{I a S} [@code{finv}] treats the expression
on top of the stack as a function of the specified variable and solves
to find the inverse function, written in terms of the same variable.
For example, @kbd{I a S x} inverts @cite{2x + 6} to @cite{x/2 - 3}.
You can use both Inverse and Hyperbolic [@code{ffinv}] to obtain a
fully general inverse, as described above.
X
@kindex a P
@pindex calc-poly-roots
@tindex roots
Some equations, specifically polynomials, have a known, finite number
of solutions.  The @kbd{a P} (@code{calc-poly-roots}) [@code{roots}]
command uses @kbd{H a S} to solve an equation in general form, then, for
all arbitrary-sign variables like @code{s1}, and all arbitrary-integer
variables like @code{n1} for which @code{n1} only usefully varies over
a finite range, it expands these variables out to all their possible
values.  The results are collected into a vector, which is returned.
For example, @samp{roots(x^4 = 1, x)} returns the four solutions
@samp{[1, -1, (0, 1), (0, -1)]}.  Generally an @var{n}th degree
polynomial will always have @var{n} roots on the complex plane.
(If you have given a @code{real} declaration for the solution
variable, then only the real-valued solutions, if any, will be
reported; @pxref{Declarations}.)
X
Note that because @kbd{a P} uses @kbd{H a S}, it is able to deliver
symbolic solutions if the polynomial has symbolic coefficients.  Also
note that Calc's solver is not able to get exact symbolic solutions
to all polynomials.  Polynomials containing powers up to @cite{x^4}
can always be solved exactly; polynomials of higher degree sometimes
can be:  @cite{x^6 + x^3 + 1} is converted to @cite{(x^3)^2 + (x^3) + 1},
which can be solved for @cite{x^3} using the quadratic equation, and then
for @cite{x} by taking cube roots.  But in many cases, like
@cite{x^6 + x + 1}, Calc does not know how to rewrite the polynomial
into a form it can solve.  The @kbd{a P} command can still deliver a
list of numerical roots, however, provided that symbolic mode (@kbd{m s})
is not turned on.  (If you work with symbolic mode on, recall that the
@kbd{N} (@code{calc-eval-num}) key is a handy way to reevaluate the
formula on the stack with symbolic mode temporarily off.)  Naturally,
@kbd{a P} can only provide numerical roots if the polynomial coefficents
are all numbers (real or complex).
X
@node Solving Systems of Equations, Decomposing Polynomials, Multiple Solutions, Solving Equations
@subsection Solving Systems of Equations
X
@noindent
@cindex Systems of equations, symbolic
You can also use the commands described above to solve systems of
simultaneous equations.  Just create a vector of equations, then
specify a vector of variables for which to solve.  (You can omit
the surrounding brackets when entering the vector of variables
at the prompt.)
X
For example, putting @samp{[x + y = a, x - y = b]} on the stack
and typing @kbd{a S x,y @key{RET}} produces the vector of solutions
@samp{[x = a - (a-b)/2, y = (a-b)/2]}.  The result vector will
have the same length as the variables vector, and the variables
will be listed in the same order there.  Note that the solutions
are not always simplified as far as possible; the solution for
@cite{x} here could be improved by an application of @kbd{j M}
(@code{calc-sel-merge}) on the @samp{/} sign, followed by
@kbd{M-2 j v} also on the @samp{/} sign (which runs @kbd{a s},
@code{calc-simplify}, to simplify the quotient).
X
Calc's algorithm works by trying to eliminate one variable at a
time by solving one of the equations for that variable and then
substituting into the other equations.  Calc will try all the
possiblities, but you can speed things up by noting that Calc
first tries to eliminate the first variable with the first
equation, then the second variable with the second equation,
and so on.  It also helps to put the simpler (e.g., more linear)
equations toward the front of the list.  Calc's algorithm will
solve any system of linear equations, and also many kinds of
nonlinear systems.
X
@tindex elim
Normally there will be as many variables as equations.  If you
give fewer variables than equations (an ``over-determined'' system
of equations), Calc will find a partial solution.  For example,
typing @kbd{a S y @key{RET}} with the above system of equations
would produce @samp{[y = a - x]}.  There are now several ways to
express this solution in terms of the original variables; Calc uses
the first one that it finds.  You can control the choice by adding
variable specifiers of the form @samp{elim(@var{v})} to the
variables list.  This says that @var{v} should be eliminated from
the equations; the variable will not appear at all in the solution.
For example, typing @kbd{a S y,elim(x)} would yield
@samp{[y = a - (b+a)/2]}.
X
If the variables list contains only @code{elim} specifiers,
Calc simply eliminates those variables from the equations
and then returns the resulting equations.  For example,
@kbd{a S elim(x)} produces @samp{[a - 2 y = b]}.  Every variable
eliminated will reduce the number of equations in the system
by one.
X
Again, @kbd{a S} gives you one solution to the system of
equations.  If there are several solutions, you can use @kbd{H a S}
to get a general family of solutions, or, if there is a finite
number of solutions, you can use @kbd{a P} to get a list.  (In
the latter case, the result will take the form of a matrix where
the rows are different solutions and the columns correspond to the
variables you requested.)
X
Another way to deal with certain kinds of overdetermined systems of
equations is the @kbd{a F} command, which does least-squares fitting
to satisfy the equations.  @xref{Curve Fitting}.
X
@node Decomposing Polynomials, , Solving Systems of Equations, Solving Equations
@subsection Decomposing Polynomials
X
@noindent
@tindex poly
The @code{poly} function takes a polynomial and a variable as
arguments, and returns a vector of polynomial coefficients (constant
coefficient first).  For example, @samp{poly(x^3 + 2 x, x)} returns
@cite{[0, 2, 0, 1]}.  If the input is not a polynomial in @cite{x},
the call to @code{poly} is left in symbolic form.  If the input does
not involve the variable @cite{x}, the input is returned in a list
of length one, representing a polynomial with only a constant
coefficient.  The call @samp{poly(x, x)} returns the vector @cite{[0, 1]}.
The last element of the returned vector is guaranteed to be nonzero;
note that @samp{poly(0, x)} returns the empty vector @cite{[]}.
Note that @cite{x} may actually be any formula; for example,
@samp{poly(sin(x)^2 - sin(x) + 3, sin(x))} returns @cite{[3, -1, 1]}.
X
@cindex Coefficients of polynomial
@cindex Degree of polynomial
To get the @cite{x^k} coefficient of polynomial @cite{p}, use
@samp{poly(p, x)_(k+1)}.  To get the degree of polynomial @cite{p},
use @samp{vlen(poly(p, x)) - 1}.  For example, @samp{poly((x+1)^4, x)}
returns @samp{[1, 4, 6, 4, 1]}, so @samp{poly((x+1)^4, x)_(2+1)}
gives the @cite{x^2} coefficient of this polynomial, 6.
X
@tindex gpoly
One important feature of the solver is its ability to recognize
formulas which are ``essentially'' polynomials.  This ability is
made available to the user through the @code{gpoly} function, which
is used just like @code{poly}:  @samp{gpoly(@var{expr}, @var{var})}.
If @var{expr} is a polynomial in some term which includes @var{var}, then
this function will return a vector @samp{[@var{x}, @var{c}, @var{a}]}
where @var{x} is the term that depends on @var{var}, @var{c} is a
vector of polynomial coefficients (like the one returned by @code{poly}),
and @var{a} is a multiplier which is usually 1.  Basically,
@samp{@var{expr} = @var{a}*(@var{c}_1 + @var{c}_2 @var{x} +
@var{c}_3 @var{x}^2 + ...)}.  The last element of @var{c} is
guaranteed to be non-zero, and @var{c} will not equal @samp{[1]}
(i.e., the trivial decomposition @var{expr} = @var{x} is not
considered a polynomial).  One side effect is that @samp{gpoly(x, x)}
and @samp{gpoly(6, x)}, both of which might be expected to recognize
their arguments as polynomials, will not because the decomposition
is considered trivial.
X
For example, @samp{gpoly((x-2)^2, x)} returns @samp{[x, [4, -4, 1], 1]},
since the expanded form of this polynomial is @cite{4 - 4 x + x^2}.
X
The term @var{x} may itself be a polynomial in @var{var}.  This is
done to reduce the size of the @var{c} vector.  For example,
@samp{gpoly(x^4 + x^2 - 1, x)} returns @samp{[x^2, [-1, 1, 1], 1]},
since a quadratic polynomial in @cite{x^2} is easier to solve than
a quartic polynomial in @cite{x}.
X
A few more examples of the kinds of polynomials @code{gpoly} can
discover:
X
@smallexample
sin(x) - 1               [sin(x), [-1, 1], 1]
x + 1/x - 1              [x, [1, -1, 1], 1/x]
x + 1/x                  [x^2, [1, 1], 1/x]
x^3 + 2 x                [x^2, [2, 1], x]
x + x^2:3 + sqrt(x)      [x^1:6, [1, 1, 0, 1], x^1:2]
x^(2a) + 2 x^a + 5       [x^a, [5, 2, 1], 1]
(exp(-x) + exp(x)) / 2   [e^(2 x), [0.5, 0.5], e^-x]
@end smallexample
X
The @code{poly} and @code{gpoly} functions accept a third integer argument
which specifies the largest degree of polynomial that is acceptable.
If this is @cite{n}, then only @var{c} vectors of length @cite{n+1}
or less will be returned.  Otherwise, the @code{poly} or @code{gpoly}
call will remain in symbolic form.  For example, the equation solver
can handle quartics and smaller polynomials, so it calls
@samp{gpoly(@var{expr}, @var{var}, 4)} to discover whether @var{expr}
can be treated by its linear, quadratic, cubic, or quartic formulas.
X
@tindex pdeg
The @code{pdeg} function computes the degree of a polynomial;
@samp{pdeg(p,x)} is the highest power of @code{x} that appears in
@code{p}.  This is the same as @samp{vlen(poly(p,x))-1}, but is
much more efficient.  If @code{p} is constant with respect to @code{x},
then @samp{pdeg(p,x) = 0}.  If @code{p} is not a polynomial in @code{x}
(e.g., @samp{pdeg(2 cos(x), x)}, the function remains unevaluated.
It is possible to omit the second argument @code{x}, in which case
@samp{pdeg(p)} returns the highest total degree of any term of the
polynomial, counting all variables that appear in @code{p}.  Note
that @code{pdeg(c) = pdeg(c,x) = 0} for any nonzero constant @code{c};
the degree of the constant zero is considered to be @code{-inf}
(minus infinity).
X
@tindex plead
The @code{plead} function finds the leading term of a polynomial.
Thus @samp{plead(p,x)} is equivalent to @samp{poly(p,x)_vlen(poly(p,x))},
though again more efficient.  In particular, @samp{plead((2x+1)^10, x)}
returns 1024 without expanding out the list of coefficients.  The
value of @code{plead(p,x)} will be zero only if @cite{p = 0}.
X
@tindex pcont
The @code{pcont} function finds the @dfn{content} of a polynomial.  This
is the greatest common divisor of all the coefficients of the polynomial.
With two arguments, @code{pcont(p,x)} effectively uses @samp{poly(p,x)}
to get a list of coefficients, then uses @code{pgcd} (the polynomial
GCD function) to combine these into an answer.  For example,
@samp{pcont(4 x y^2 + 6 x^2 y, x)} is @samp{2 y}.  The content is
basically the ``biggest'' polynomial that can be divided into @code{p}
exactly.  The sign of the content is the same as the sign of the leading
coefficient.
X
With only one argument, @samp{pcont(p)} computes the numerical
content of the polynomial, i.e., the @code{gcd} of the numerical
coefficients of all the terms in the formula.  Note that @code{gcd}
is defined on rational numbers as well as integers; it computes
the @code{gcd} of the numerators and the @code{lcm} of the
denominators.  Thus @samp{pcont(4:3 x y^2 + 6 x^2 y)} returns 2:3.
Dividing the polynomial by this number will clear all the
denominators, as well as dividing by any common content in the
numerators.  The numerical content of a polynomial is negative only
if all the coefficients in the polynomial are negative.
X
@tindex pprim
The @code{pprim} function finds the @dfn{primitive part} of a
polynomial, which is simply the polynomial divided (using @code{pdiv}
if necessary) by its content.  If the input polynomial has rational
coefficients, the result will have integer coefficients in simplest
terms.
X
@node Numerical Solutions, Curve Fitting, Solving Equations, Algebra
@section Numerical Solutions
X
@noindent
Not all equations can be solved symbolically.  The commands in this
section use numerical algorithms that can find a solution to a specific
instance of an equation to any desired accuracy.  Note that these
commands are slower than their algebraic cousins; it is a good idea
to try @kbd{a S} before resorting to @kbd{a R}.
X
(@xref{Curve Fitting}, for some other, more specialized, operations
on numerical data.)
X
@menu
* Root Finding::
* Minimization::
* Numerical Systems of Equations::
@end menu
X
@node Root Finding, Minimization, Numerical Solutions, Numerical Solutions
@subsection Root Finding
X
@noindent
@kindex a R
@pindex calc-find-root
@tindex root
@cindex Newton's method
@cindex Roots of equations
@cindex Numerical root-finding
The @kbd{a R} (@code{calc-find-root}) [@code{root}] command finds a
numerical solution (or @dfn{root}) of an equation.  (This command treats
inequalities the same as equations.  If the input is any other kind
of formula, it is interpreted as an equation of the form @cite{X = 0}.)
X
The @kbd{a R} command requires an initial guess on the top of the
stack, and a formula in the second-to-top position.  It prompts for a
solution variable, which must appear in the formula.  All other variables
that appear in the formula must have assigned values, i.e., when
a value is assigned to the solution variable and the formula is
evaluated with @kbd{=}, it should evaluate to a number.  Any assigned
value for the solution variable itself is ignored and unaffected by
this command.
X
When the command completes, the initial guess is replaced on the stack
by a vector of two numbers:  The value of the solution variable that
solves the equation, and the difference between the lefthand and
righthand sides of the equation at that value.  Ordinarily, the second
number will be zero or very nearly zero.  (Note that Calc uses a
slightly higher precision while finding the root, and thus the second
number may be slightly different from the value you would compute from
the equation yourself.)
X
The @kbd{v h} (@code{calc-head}) command is a handy way to extract
the first element of the result vector, discarding the error term.
X
The initial guess can be a real number, in which case Calc searches
for a real solution near that number, or a complex number, in which
case Calc searches the whole complex plane near that number for a
solution, or it can be an interval form which restricts the search
to real numbers inside that interval.
X
Calc tries to use @kbd{a d} to take the derivative of the equation.
If this succeeds, it uses Newton's method.  If the equation is not
differentiable Calc uses a bisection method.  (If Newton's method
appears to be going astray, Calc switches over to bisection if it
can, or otherwise gives up.  In this case it may help to try again
with a slightly different initial guess.)  If the initial guess is a
complex number, the function must be differentiable.
X
If the formula (or the difference between the sides of an equation)
is negative at one end of the interval you specify and positive at
the other end, the root finder is guaranteed to find a root.
Otherwise, Calc subdivides the interval into small parts looking for
positive and negative values to bracket the root.  When your guess is
an interval, Calc will not look outside that interval for a root.
X
@kindex H a R
@tindex wroot
The @kbd{H a R} [@code{wroot}] command is similar to @kbd{a R}, except
that if the initial guess is an interval for which the function has
the same sign at both ends, then rather than subdividing the interval
Calc attempts to widen it to enclose a root.  Use this mode if
you are not sure if the function has a root in your interval.
X
If the function is not differentiable, and you give a simple number
instead of an interval as your initial guess, Calc uses this widening
process even if you did not type the Hyperbolic flag.  (If the function
@emph{is} differentiable, Calc uses Newton's method which does not
require a bounding interval in order to work.)
X
If Calc leaves the @code{root} or @code{wroot} function in symbolic
form on the stack, it will normally display an explanation for why
no root was found.  If you miss this explanation, press @kbd{w}
(@code{calc-why}) to get it back.
X
@node Minimization, Numerical Systems of Equations, Root Finding, Numerical Solutions
@subsection Minimization
X
@noindent
@kindex a N
@kindex H a N
@kindex a X
@kindex H a X
@pindex calc-find-minimum
@pindex calc-find-maximum
@tindex minimize
@tindex maximize
@cindex Minimization, numerical
The @kbd{a N} (@code{calc-find-minimum}) [@code{minimize}] command
finds a minimum value for a formula.  It is very similar in operation
to @kbd{a R} (@code{calc-find-root}):  You give the formula and an initial
guess on the stack, and are prompted for the name of a variable.  The guess
may be either a number near the desired minimum, or an interval enclosing
the desired minimum.  The function returns a vector containing the
value of the the variable which minimizes the formula's value, along
with the minimum value itself.
X
Note that this command looks for a @emph{local} minimum.  Many functions
have more than one minimum; some, like @c{$x \sin x$}
@cite{x sin(x)}, have infinitely
many.  In fact, there is no easy way to define the ``global'' minimum
of @c{$x \sin x$}
@cite{x sin(x)} but Calc can still locate any particular local minimum
for you.  Calc basically goes downhill from the initial guess until it
finds a point at which the function's value is greater both to the left
and to the right.  Calc does not use derivatives when minimizing a function.
X
If your initial guess is an interval and it looks like the minimum
occurs at one or the other endpoint of the interval, Calc will return
that endpoint only if that endpoint is closed; thus, minimizing @cite{17 x}
over @cite{[2..3]} will return @cite{[2, 38]}, but minimizing over
@cite{(2..3]} would report no minimum found.
X
Most functions are smooth and flat near their minimum values.  Because
of this flatness, if the current precision is, say, 12 digits, the
variable can only be determined meaningfully to about six digits.  Thus
you should set the precision to twice as many digits as you need in your
answer.
X
@tindex wminimize
@tindex wmaximize
The @kbd{H a N} [@code{wminimize}] command, analogously to @kbd{H a R},
expands the guess interval to enclose a minimum rather than requiring
that the minimum lie inside the interval you supply.
X
The @kbd{a X} (@code{calc-find-maximum}) [@code{maximize}] and
@kbd{H a X} [@code{wmaximize}] commands effectively minimize the
negative of the formula you supply.
X
The formula must evaluate to a real number at all points inside the
interval (or near the initial guess if the guess is a number).  If
the initial guess is a complex number the variable will be minimized
over the complex numbers; if it is real or an interval it will
be minimized over the reals.
X
@node Numerical Systems of Equations, , Minimization, Numerical Solutions
@subsection Systems of Equations
X
@noindent
@cindex Systems of equations, numerical
The @kbd{a R} command can also solve systems of equations.  In this
case, the equation should instead be a vector of equations, the
guess should instead be a vector of numbers (intervals are not
supported), and the variable should be a vector of variables.  You
can omit the brackets while entering the list of variables.  Each
equation must be differentiable by each variable for this mode to
work.  The result will be a vector of two vectors:  The variable
values that solved the system of equations, and the differences
between the sides of the equations with those variable values.
There must be the same number of equations as variables.  Since
only plain numbers are allowed as guesses, the Hyperbolic flag has
no effect when solving a system of equations.
X
It is also possible to minimize over many variables with @kbd{a N}
(or maximize with @kbd{a X}).  Once again the variable name should
be replaced by a vector of variables, and the initial guess should
be an equal-sized vector of initial guesses.  But, unlike the case of
multidimensional root finding, the formula being minimized should
still be a single formula, @emph{not} a vector.  Beware that
multidimensional minimization is currently @emph{very} slow.
X
@node Curve Fitting, Summations, Numerical Solutions, Algebra
@section Curve Fitting
X
@noindent
The @kbd{a F} command fits a set of data to a @dfn{model formula},
such as @cite{y = m x + b} where @cite{m} and @cite{b} are parameters
to be determined.  For a typical set of measured data there will be
no single @cite{m} and @cite{b} that exactly fit the data; in this
case, Calc chooses values of the parameters that provide the closest
possible fit.
X
@menu
* Linear Fits::
* Polynomial and Multilinear Fits::
* Error Estimates for Fits::
* Standard Nonlinear Models::
* Curve Fitting Details::
* Interpolation::
@end menu
X
@node Linear Fits, Polynomial and Multilinear Fits, Curve Fitting, Curve Fitting
@subsection Linear Fits
X
@noindent
@kindex a F
@pindex calc-curve-fit
@tindex fit
@cindex Linear regression
@cindex Least-squares fits
The @kbd{a F} (@code{calc-curve-fit}) [@code{fit}] command attempts
to fit a set of data (@cite{x} and @cite{y} vectors of numbers) to a
straight line, polynomial, or other function of @cite{x}.  For the
moment we will consider only the case of fitting to a line, and we
will ignore the issue of whether or not the model was in fact a good
fit for the data.
X
In a standard linear least-squares fit, we have a set of @cite{(x,y)}
data points that we wish to fit to the @dfn{model} @cite{y = m x + b}
by adjusting the parameters @cite{m} and @cite{b} to make the calculated
@cite{y} values from the formula as close as possible to the actual
@cite{y} values from the data set.  (In a polynomial fit, the model is
instead, say, @cite{y = a x^3 + b x^2 + c x + d}.  In a multilinear fit,
we have data points of the form @cite{(x1,x2,x3,y)} and our model is
@cite{y = a x1 + b x2 + c x3 + d}.  These will be discussed later.)
X
In the model formula, variables like @cite{x} and @cite{x2} are called
the @dfn{independent variables}, and @cite{y} is the @dfn{dependent
variable}.  Variables like @cite{m}, @cite{a}, and @cite{b} are called
the @dfn{parameters} of the model.
X
The @kbd{a F} command takes the data set to be fitted from the stack.
By default, it expects the data in the form of a matrix.  For example,
for a linear or polynomial fit, this would be a @c{$2\times N$}
@asis{2xN} matrix where
the first row is a list of @cite{x} values and the second row has the
corresponding @cite{y} values.  For the multilinear fit shown above,
the matrix would have four rows (@cite{x1}, @cite{x2}, @cite{x3}, and
@cite{y}, respectively).
X
If you happen to have an @c{$N\times2$}
@asis{Nx2} matrix instead of a @c{$2\times N$}
@asis{2xN} matrix,
just press @kbd{v t} first to transpose the matrix.
X
After you type @kbd{a F}, Calc prompts you to select a model.  For a
linear fit, press the digit @kbd{1}.
X
Calc then prompts for you to name the variables.  By default it chooses
high letters like @cite{x} and @cite{y} for independent variables and
low letters like @cite{a} and @cite{b} for parameters.  (The dependent
variable doesn't need a name.)  The two kinds of variables are separated
by a semicolon.  Since you generally care more about the names of the
independent variables than of the parameters, Calc also allows you to
name only them and let the parameters use default names.
X
For example, suppose the data matrix
X
@ifinfo
@group
@example
[ [ 1, 2, 3, 4,  5  ]
X  [ 5, 7, 9, 11, 13 ] ]
@end example
@end group
@end ifinfo
@tex
\turnoffactive
\turnoffactive
$$ \pmatrix{ 1 & 2 & 3 & 4  & 5  \cr
X             5 & 7 & 9 & 11 & 13 }
$$
@end tex
X
@noindent
is on the stack and we wish to do a simple linear fit.  Type
@kbd{a F}, then @kbd{1} for the model, then @kbd{RET} to use
the default names.  The result will be the formula @cite{3 + 2 x}
on the stack.  Calc has created the model expression @kbd{a + b x},
then found the optimal values of @cite{a} and @cite{b} to fit the
data.  (In this case, it was able to find an exact fit.)  Calc then
substituted those values for @cite{a} and @cite{b} in the model.
X
The @kbd{a F} command puts two entries in the trail.  One is, as
always, a copy of the result that went to the stack; the other is
a vector of the actual parameter values, written as equations:
@cite{[a = 3, b = 2]}, in case you'd rather read them from a
vector than pick them out of the formula.  (You can type @kbd{t y}
to move this vector to the stack; @pxref{Trail Commands}.)
X
Specifying a different independent variable name will affect the
resulting formula: @kbd{a F 1 k RET} produces @kbd{3 + 2 k}.
Changing the parameter names (@kbd{a F 1 k;b,m RET}) will affect
the equations that go into the trail.
X
@tex
\bigskip
@end tex
X
To see what happens when the fit is not exact, we could change
the number 13 in the data matrix to 14 and try the fit again.
The result is:
X
@example
2.6 + 2.2 x
@end example
X
Evaluating this formula, say with @kbd{v x 5 RET TAB V M $ RET}, shows
a reasonably close match to the y-values in the data.
X
@example
[4.8, 7., 9.2, 11.4, 13.6]
@end example
X
Since there is no line which passes through all the @i{N} data points,
Calc has chosen a line that best approximates the data points using
the method of least squares.  The idea is to define the @dfn{chi-square}
error measure
X
@ifinfo
@example
chi^2 = sum((y_i - (a + b x_i))^2, i, 1, N)
@end example
@end ifinfo
@tex
\turnoffactive
$$ \chi^2 = \sum_{i=1}^N (y_i - (a + b x_i))^2 $$
@end tex
X
@noindent
which is clearly zero if @cite{a + b x} exactly fits all data points,
and increases as various @cite{a + b x_i} values fail to match the
corresponding @cite{y_i} values.  There are several reasons why the
summand is squared, one of them being to ensure that @c{$\chi^2 \ge 0$}
@cite{chi^2 >= 0}.
Least-squares fitting simply chooses the values of @cite{a} and @cite{b}
for which the error @c{$\chi^2$}
@cite{chi^2} is as small as possible.
X
Other kinds of models do the same thing but with a different model
formula in place of @cite{a + b x_i}.
X
@tex
\bigskip
@end tex
X
A numeric prefix argument causes the @kbd{a F} command to take the
data in some other form than one big matrix.  A positive argument @i{N}
will take @i{N} items from the stack, corresponding to the @i{N} rows
of a data matrix.  In the linear case, @i{N} must be 2 since there
is always one independent variable and one dependent variable.
X
A prefix of zero or plain @kbd{C-u} is a compromise; Calc takes two
items from the stack, an @i{N}-row matrix of @cite{x} values, and a
vector of @cite{y} values.  If there is only one independent variable,
the @cite{x} values can be either a one-row matrix or a plain vector,
in which case the @kbd{C-u} prefix is the same as a @kbd{C-u 2} prefix.
SHAR_EOF
true || echo 'restore of calc.texinfo failed'
fi
echo 'End of  part 46'
echo 'File calc.texinfo is continued in part 47'
echo 47 > _shar_seq_.tmp
exit 0
exit 0 # Just in case...
--
Kent Landfield                   INTERNET: [e-mail]
Sterling Software, IMD           UUCP:     uunet!sparky!kent
Phone:    [phone]         FAX:      [phone]
Please send comp.sources.misc-related mail to [e-mail].