calc/cal/README

Calc standard resource files
----------------------------

To load a resource file, try:

    read filename

You do not need to add the .cal extension to the filename.  Calc
will search along the $CALCPATH (see ``help environment'').

Normally a resource file will simply define some functions.  By default,
most resource files will print out a short message when they are read.
For example:

    ; read lucas
    lucas(h,n) defined
    gen_u2(h,n,v1) defined
    gen_u0(h,n,v1) defined
    rodseth_xhn(x,h,n) defined
    gen_v1(h,n) defined
    ldebug(funct,str) defined
    legacy_gen_v1(h,n) defined

will cause calc to load and execute the 'lucas.cal' resource file.
Executing the resource file will cause several functions to be defined.
Executing the lucas function:

    ; lucas(149,60)
	    1
    ; lucas(146,61)
	    0

shows that 149*2^60-1 is prime whereas 146*2^61-1 is not.

=-=

Calc resource file files are provided because they serve as examples of
how use the calc language, and/or because the authors thought them to
be useful!

If you write something that you think is useful, please join the
low volume calc mailing list calc-tester.  Then send your contribution
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=-=

By convention, a resource file only defines and/or initializes functions,
objects and variables.	(The regress.cal and testxxx.cal regression test
suite is an exception.)	 Also by convention, an additional usage message
regarding important object and functions is printed.

If a resource file needs to load another resource file, it should use
the -once version of read:

    /* pull in needed resource files */
    read -once "surd"
    read -once "lucas"

This will cause the needed resource files to be read once.  If these
files have already been read, the read -once will act as a noop.

The "resource_debug" parameter is intended for controlling the possible
display of special information relating to functions, objects, and
other structures created by instructions in calc resource files.
Zero value of config("resource_debug") means that no such information
is displayed.  For other values, the non-zero bits which currently
have meanings are as follows:

    n		Meaning of bit n of config("resource_debug")

    0	When a function is defined, redefined or undefined at
	interactive level, a message saying what has been done
	is displayed.

    1	When a function is defined, redefined or undefined during
	the reading of a file, a message saying what has been done
	is displayed.

    2	Show func will display more information about a functions
	arguments as well as more argument summary information.

    3	During execution, allow calc standard resource files
	to output additional debugging information.

The value for config("resource_debug") in both oldstd and newstd is 3,
but if calc is invoked with the -d flag, its initial value is zero.
Thus, if calc is started without the -d flag, until config("resource_debug")
is changed, a message will be output when a function is defined
either interactively or during the reading of a file.

Sometimes the information printed is not enough.  In addition to the
standard information, one might want to print:

	* useful obj definitions
	* functions with optional args
	* functions with optional args where the param() interface is used

For these cases we suggest that you place at the bottom of your code
something that prints extra information if config("resource_debug") has
either of the bottom 2 bits set:

	if (config("resource_debug") & 3) {
		print "obj xyz defined";
		print "funcA([val1 [, val2]]) defined";
		print "funcB(size, mass, ...) defined";
	}

If your the resource file needs to output special debugging information,
we recommend that you check for bit 3 of the config("resource_debug")
before printing the debug statement:

	if (config("resource_debug") & 8) {
		print "DEBUG: This a sample debug statement";
	}

=-=

The following is a brief description of some of the calc resource files
that are shipped with calc.  See above for example of how to read in
and execute these files.

alg_config.cal

    global test_time
    mul_loop(repeat,x) defined
    mul_ratio(len) defined
    best_mul2() defined
    sq_loop(repeat,x) defined
    sq_ratio(len) defined
    best_sq2() defined
    pow_loop(repeat,x,ex) defined
    pow_ratio(len) defined
    best_pow2() defined

    These functions search for an optimal value of config("mul2"),
    config("sq2"), and config("pow2").  The calc default values of these
    configuration values were set by running this resource file on a
    1.8GHz AMD 32-bit CPU of ~3406 BogoMIPS.

    The best_mul2() function returns the optimal value of config("mul2").
    The best_sq2() function returns the optimal value of config("sq2").
    The best_pow2() function returns the optimal value of config("pow2").
    The other functions are just support functions.

    By design, best_mul2(), best_sq2(), and best_pow2() take a few
    minutes to run.  These functions increase the number of times a
    given computational loop is executed until a minimum amount of CPU
    time is consumed.  To watch these functions progress, one can set
    the config("user_debug") value.

    Here is a suggested way to use this resource file:

	; read alg_config
	; config("user_debug",2),;
	; best_mul2(); best_sq2(); best_pow2();
	; best_mul2(); best_sq2(); best_pow2();
	; best_mul2(); best_sq2(); best_pow2();

    NOTE: It is perfectly normal for the optimal value returned to differ
    slightly from run to run.  Slight variations due to inaccuracy in
    CPU timings will cause the best value returned to differ slightly
    from run to run.

    One can use a calc startup file to change the initial values of
    config("mul2"), config("sq2"), and config("pow2").  For example one
    can place into ~/.calcrc these lines:

	config("mul2", 1780),;
	config("sq2", 3388),;
	config("pow2", 176),;

    to automatically and silently change these config values.
    See help/config and CALCRC in help/environment for more information.


beer.cal

    This calc resource is calc's contribution to the 99 Bottles of Beer
    web page:

	http://www.ionet.net/~timtroyr/funhouse/beer.html#calc

     NOTE: This resource produces a lot of output.  :-)


bernoulli.cal

    B(n)

    Calculate the nth Bernoulli number.

    NOTE: There is now a bernoulli() builtin function.  This file is
    	  left here for backward compatibility and now simply returns
	  the builtin function.


bernpoly.cal

    bernpoly(n,z)

    Computes the nth Bernoulli polynomial at z for arbitrary n,z.  See:

        http://en.wikipedia.org/wiki/Bernoulli_polynomials
        http://mathworld.wolfram.com/BernoulliPolynomial.html

    for further information


bigprime.cal

    bigprime(a, m, p)

    A prime test, base a, on p*2^x+1 for even x>m.


brentsolve.cal

    brentsolve(low, high,eps)

    A root-finder implementwed with the Brent-Dekker trick.

    brentsolve2(low, high,which,eps)

    The second function, brentsolve2(low, high,which,eps) has some lines
    added to make it easier to hardcode the name of the helper function
    different from the obligatory "f".

    See:

        http://en.wikipedia.org/wiki/Brent%27s_method
        http://mathworld.wolfram.com/BrentsMethod.html

    to find out more about the Brent-Dekker method.


constants.cal

    e()
    G()

    An implementation of different constants to arbitrary precision.


chi.cal

    Z(x[, eps])
    P(x[, eps])
    chi_prob(chi_sq, v[, eps])

    Computes the Probability, given the Null Hypothesis, that a given
    Chi squared values >= chi_sq with v degrees of freedom.

    The chi_prob() function does not work well with odd degrees of freedom.
    It is reasonable with even degrees of freedom, although one must give
    a sufficiently small error term as the degrees gets large (>100).

    The Z(x) and P(x) are internal statistical functions.

    eps is an optional epsilon() like error term.


chrem.cal

    chrem(r1,m1 [,r2,m2, ...])
    chrem(rlist, mlist)

    Chinese remainder theorem/problem solver.


deg.cal

    deg(deg, min, sec)
    deg_add(a, b)
    deg_neg(a)
    deg_sub(a, b)
    deg_mul(a, b)
    deg_print(a)

    Calculate in degrees, minutes, and seconds.  For a more functional
    version see dms.cal.


dms.cal

    dms(deg, min, sec)
    dms_add(a, b)
    dms_neg(a)
    dms_sub(a, b)
    dms_mul(a, b)
    dms_print(a)
    dms_abs(a)
    dms_norm(a)
    dms_test(a)
    dms_int(a)
    dms_frac(a)
    dms_rel(a,b)
    dms_cmp(a,b)
    dms_inc(a)
    dms_dec(a)

    Calculate in degrees, minutes, and seconds.  Unlike deg.cal, increments
    are on the arc second level.  See also hms.cal.


dotest.cal

    dotest(dotest_file [,dotest_code [,dotest_maxcond]])

    dotest_file

	Search along CALCPATH for dotest_file, which contains lines that
	should evaluate to 1.  Comment lines and empty lines are ignored.
	Comment lines should use ## instead of the multi like /* ... */
	because lines are evaluated one line at a time.

    dotest_code

	Assign the code number that is to be printed at the start of
	each non-error line and after **** in each error line.
	The default code number is 999.

    dotest_maxcond

	The maximum number of error conditions that may be detected.
	An error condition is not a sign of a problem, in some cases
	a line deliberately forces an error condition.	A value of -1,
	the default, implies a maximum of 2147483647.

    Global variables and functions must be declared ahead of time because
    the dotest scope of evaluation is a line at a time.  For example:

	read dotest.cal
	read set8700.cal
	dotest("set8700.line");


factorial.cal

    factorial(n)

    Calculates the product of the positive integers up to and including n.

    See:

	http://en.wikipedia.org/wiki/Factorial

    for information on the factorial. This function depends on the script
    toomcook.cal.


    primorial(a,b)

    Calculates the product of the primes between a and b. If a is not prime
    the next higher prime is taken as the starting point. If b is not prime
    the next lower prime is taking as the end point b. The end point b must
    not exceed 4294967291.  See:

	http://en.wikipedia.org/wiki/Primorial

    for information on the primorial.


factorial2.cal

    This file contents a small variety of integer functions that can, with
    more or less pressure, be related to the factorial.

    doublefactorial(n)

    Calculates the double factorial n!! with different algorithms for
        - n odd
        - n even and positive
        - n (real|complex) sans the negative half integers

    See:

        http://en.wikipedia.org/wiki/Double_factorial
        http://mathworld.wolfram.com/DoubleFactorial.html

    for information on the double factorial. This function depends on
    the script toomcook.cal, factorial.cal and specialfunctions.cal.


    binomial(n,k)

    Calculates the binomial coefficients for n large and k = k \pm
    n/2. Defaults to the built-in function for smaller and/or different
    values. Meant as a complete replacement for comb(n,k) with only a
    very small overhead.  See:

        http://en.wikipedia.org/wiki/Binomial_coefficient

    for information on the binomial. This function depends on the script
    toomcook.cal factorial.cal and specialfunctions.cal.


    bigcatalan(n)

    Calculates the n-th Catalan number for n large. It is usefull
    above n~50,000 but defaults to the builtin function for smaller
    values.Meant as a complete replacement for catalan(n) with only a
    very small overhead.  See:

        http://en.wikipedia.org/wiki/Catalan_number
        http://mathworld.wolfram.com/CatalanNumber.html

    for information on Catalan numbers. This function depends on the scripts
    toomcook.cal, factorial.cal and specialfunctions.cal.


    stirling1(n,m)

    Calculates the Stirling number of the first kind. It does so with
    building a list of all of the smaller results. It might be a good
    idea, though, to run it once for the highest n,m first if many
    Stirling  numbers are needed at once, for example in a series.  See:

        http://en.wikipedia.org/wiki/Stirling_numbers_of_the_first_kind
        http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html
        Algorithm 3.17,  Donald Kreher and Douglas Simpson, "Combinatorial
          Algorithms", CRC Press, 1998, page 89.

    for information on Stirling numbers of the first kind.


    stirling2(n,m)
    stirling2caching(n,m)

    Calculate the Stirling number of the second kind.
    The first function stirling2(n,m) does it with the sum
                       m
                      ====
                 1    \      n      m - k
                 --    >    k  (- 1)      binomial(m, k)
                 m!   /
                      ====
                      k = 0

    The other function stirling2caching(n,m) does it by way of the
    reccurence relation and keeps all earlier results. This function
    is much slower for computing a single value than stirling2(n,m) but
    is very usefull if many Stirling numbers are needed, for example in
    a series.  See:

        http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind
        http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html

        Algorithm 3.17,  Donald Kreher and Douglas Simpson, "Combinatorial
          Algorithms", CRC Press, 1998, page 89.

    for information on Stirling numbers of the second kind.


    bell(n)

    Calculate the n-th Bell number. This may take some time for large n.
    See:

        http://oeis.org/A000110
        http://en.wikipedia.org/wiki/Bell_number
        http://mathworld.wolfram.com/BellNumber.html

    for information on Bell numbers.


    subfactorial(n)

    Calculate the n-th subfactorial or derangement. This may take some
    time for large n.  See:

        http://mathworld.wolfram.com/Derangement.html
        http://en.wikipedia.org/wiki/Derangement

    for information on subfactorials.


    risingfactorial(x,n)

    Calculates the rising factorial or Pochammer symbol of almost arbitrary
    x,n.  See:

        http://en.wikipedia.org/wiki/Pochhammer_symbol
        http://mathworld.wolfram.com/PochhammerSymbol.html

    for information on rising factorials.

    fallingfactorial(x,n)

    Calculates the rising factorial of almost arbitrary x,n.  See:

        http://en.wikipedia.org/wiki/Pochhammer_symbol
        http://mathworld.wolfram.com/PochhammerSymbol.html

    for information on falling factorials.


ellip.cal

    efactor(iN, ia, B, force)

    Attempt to factor using the elliptic functions: y^2 = x^3 + a*x + b.


gvec.cal

    gvec(function, vector)

    Vectorize any single-input function or trailing operator.


hello.cal

    Calc's contribution to the Hello World! page:

	http://www.latech.edu/~acm/HelloWorld.shtml
	http://www.latech.edu/~acm/helloworld/calc.html

     NOTE: This resource produces a lot of output.  :-)


hms.cal

    hms(hour, min, sec)
    hms_add(a, b)
    hms_neg(a)
    hms_sub(a, b)
    hms_mul(a, b)
    hms_print(a)
    hms_abs(a)
    hms_norm(a)
    hms_test(a)
    hms_int(a)
    hms_frac(a)
    hms_rel(a,b)
    hms_cmp(a,b)
    hms_inc(a)
    hms_dec(a)

    Calculate in hours, minutes, and seconds.  See also dmscal.


infinities.cal

    isinfinite(x)
    iscinf(x)
    ispinf(x)
    isninf(x)
    cinf()
    ninf()
    pinf()

    The symbolic handling of infinities. Needed for intnum.cal but might be
    usefull elsewhere, too.


intfile.cal

    file2be(filename)

	Read filename and return an integer that is built from the
	octets in that file in Big Endian order.  The first octets
	of the file become the most significant bits of the integer.

    file2le(filename)

	Read filename and return an integer that is built from the
	octets in that file in Little Endian order.  The first octets
	of the file become the most significant bits of the integer.

    be2file(v, filename)

	Write the absolute value of v into filename in Big Endian order.
	The v argument must be on integer.  The most significant bits
	of the integer become the first octets of the file.

    le2file(v, filename)

	Write the absolute value of v into filename in Little Endian order.
	The v argument must be on integer.  The least significant bits
	of the integer become the last octets of the file.


intnum.cal

    quadtsdeletenodes()
    quadtscomputenodes(order, expo, eps)
    quadtscore(a, b, n)
    quadts(a, b, points)
    quadglcomputenodes(N)
    quadgldeletenodes()
    quadglcore(a, b, n)
    quadgl(a, b, points)
    quad(a, b, points = -1, method = "tanhsinh")
    makerange(start, end, steps)
    makecircle(radius, center, points)
    makeellipse(angle, a, b, center, points)
    makepoints()

    This file offers some methods for numerical integration. Implemented are
    the Gauss-Legendre and the tanh-sinh quadrature.

    All functions are usefull to some extend but the main function for
    quadrature is quad(), which is not much more than an abstraction layer.

    The main workers are quadgl() for Gauss-legendre and quadts() for the
    tanh-sinh quadrature. The limits of the integral can be anything in the
    complex plane and the extended real line. The latter means that infinite
    limits are supported by way of the smbolic infinities implemented in the
    file infinities.cal (automatically linked in by intnum.cal).

    Integration in parts and contour is supported by the "points" argument
    which takes either a number or a list. the functions starting with "make"
    allow for a less error prone use.

    The function to evaluate must have the name "f".

    Examples (shamelessly stolen from mpmath):

        ; define f(x){return sin(x);}
        f(x) defined
        ; quadts(0,pi())  -  2
	    0.00000000000000000000
        ; quadgl(0,pi())  -  2
	    0.00000000000000000000

    Sometimes rounding errors accumulate, it might be a good idea to crank up
    the working precision a notch or two.

        ; define f(x){ return exp(-x^2);}
        f(x) redefined
        ; quadts(0,pinf())  - pi()
	    0.00000000000000000000
        ; quadgl(0,pinf())  - pi()
	    0.00000000000000000001

        ; define f(x){ return exp(-x^2);}
        f(x) redefined
        ; quadgl(ninf(),pinf()) - sqrt(pi())
	    0.00000000000000000000
        ; quadts(ninf(),pinf()) - sqrt(pi())
	   -0.00000000000000000000

    Using the "points" parameter is a bit tricky

        ; define f(x){ return 1/x;  }
        f(x) redefined
        ; quadts(1,1,mat[3]={1i,-1,-1i})  -  2i*pi()
	    0.00000000000000000001i
        ; quadgl(1,1,mat[3]={1i,-1,-1i})  -  2i*pi()
	    0.00000000000000000001i

    The make* functions make it a bit simpler

        ; quadts(1,1,makepoints(1i,-1,-1i))  -  2i*pi()
	    0.00000000000000000001i
        ; quadgl(1,1,makepoints(1i,-1,-1i))  -  2i*pi()
	    0.00000000000000000001i

        ; define f(x){ return abs(sin(x));}
        f(x) redefined
        ; quadts(0,2*pi(),makepoints(pi()))  - 4
	    0.00000000000000000000
        ; quadgl(0,2*pi(),makepoints(pi()))  - 4
	    0.00000000000000000000

    The quad*core functions do not offer anything fancy but the third parameter
    controls the so called "order" which is just the number of nodes computed.
    This can be quite usefull in some circumstances.

        ; quadgldeletenodes()
        ; define f(x){ return exp(x);}
        f(x) redefined
        ; s=usertime();quadglcore(-3,3)- (exp(3)-exp(-3));e=usertime();e-s
	    0.00000000000000000001
	    2.632164
        ; s=usertime();quadglcore(-3,3)- (exp(3)-exp(-3));e=usertime();e-s
	    0.00000000000000000001
	    0.016001
        ; quadgldeletenodes()
        ; s=usertime();quadglcore(-3,3,14)- (exp(3)-exp(-3));e=usertime();e-s
	   -0.00000000000000000000
	    0.024001
        ; s=usertime();quadglcore(-3,3,14)- (exp(3)-exp(-3));e=usertime();e-s
	   -0.00000000000000000000
	    0

    It is not much but can sum up. The tanh-sinh algorithm is not optimizable
    as much as the Gauss-Legendre algorithm but is per se much faster.

        ; s=usertime();quadtscore(-3,3)- (exp(3)-exp(-3));e=usertime();e-s
	    -0.00000000000000000001
	     0.128008
        ; s=usertime();quadtscore(-3,3)- (exp(3)-exp(-3));e=usertime();e-s
	    -0.00000000000000000001
	     0.036002
        ; s=usertime();quadtscore(-3,3,49)- (exp(3)-exp(-3));e=usertime();e-s
	    -0.00000000000000000000
	     0.036002
        ; s=usertime();quadtscore(-3,3,49)- (exp(3)-exp(-3));e=usertime();e-s
	    -0.00000000000000000000
	     0.01200


lambertw.cal

     lambertw(z,branch)

     Computes Lambert's W-function at "z" at branch "branch". See

         http://en.wikipedia.org/wiki/Lambert_W_function
         http://mathworld.wolfram.com/LambertW-Function.html
         https://cs.uwaterloo.ca/research/tr/1993/03/W.pdf
         http://arxiv.org/abs/1003.1628

     to get more information.

     This file includes also an implementation for the series described in
     Corless et al. (1996) eq. 4.22 (W-pdf) and Verebic (2010) (arxive link)
     eqs.35-37.

     The series has been implemented to get a different algorithm
     for checking the results. This was necessary because the results
     of the implementation in Maxima, the only program with a general
     lambert-w implementation at hand at that time, differed slightly. The
     Maxima versions tested were: Maxima 5.21.1 and 5.29.1. The current
     version of this code concurs with the results of Mathematica`s(tm)
     ProductLog[branch,z] with the tested values.

     The series is only valid for the branches 0,-1, real z, converges
     for values of z _very_ near the branchpoint -exp(-1) only, and must
     be given the branches explicitly.  See the code in lambertw.cal
     for further information.


linear.cal

    linear(x0, y0, x1, y1, x)

    Returns the value y such that (x,y) in on the line (x0,y0), (x1,y1).
    Requires x0 != y0.


lnseries.cal

    lnseries(limit)
    lnfromseries(n)
    deletelnseries()

    Calculates a series of n natural logarithms at 1,2,3,4...n. It
    does so by computing the prime factorization of all of the number
    sequence 1,2,3...n, calculates the natural logarithms of the primes
    in 1,2,3...n and uses the above factorization to build the natural
    logarithms of the rest of the sequence by sadding the logarithms of
    the primes in the factorization.  This is faster for high precision
    of the logarithms and/or long sequences.

    The sequence need to be initiated by running either lnseries(n) or
    lnfromseries(n) once with n the upper limit of the sequence.


lucas.cal

    lucas(h, n)

    Perform a primality test of h*2^n-1.

    gen_u2(h, n, v1)

    Generate u(2) for h*2^n-1.  This function is used by lucas(h, n),
    as the first term in the lucas sequence that is needed to
    prove that h*2^n-1 is prime or not prime.

    NOTE: Some call this term u(0).  The function gen_u0(h, n, v1)
    	  simply calls gen_u2(h, n, v1) for such people.  :-)

    gen_v1(h, v)

    Generate v(1) for h*2^n-1.  This function is used by lucas(h, n),
    via the gen_u2(h, n, v1), to supply the 3rd argument to gen_u2.

    legacy_gen_v1(h, n)

    Generate v(1) for h*2^n-1 using the legacy Amdahl 6 method.
    This function sometimes returns -1 for a few cases when
    h is a multiple of 3.  This function is NOT used by lucas(h, n).


lucas_chk.cal

    lucas_chk(high_n)

    Test all primes of the form h*2^n-1, with 1<=h<200 and n <= high_n.
    Requires lucas.cal to be loaded.  The highest useful high_n is 1000.

    Used by regress.cal during the 2100 test set.


mersenne.cal

    mersenne(p)

    Perform a primality test of 2^p-1, for prime p>1.


mfactor.cal

    mfactor(n [, start_k=1 [, rept_loop=10000 [, p_elim=17]]])

    Return the lowest factor of 2^n-1, for n > 0.  Starts looking for factors
    at 2*start_k*n+1.  Skips values that are multiples of primes <= p_elim.
    By default, start_k == 1, rept_loop = 10000 and p_elim = 17.

    The p_elim == 17 overhead takes ~3 minutes on an 200 Mhz r4k CPU and
    requires about ~13 Megs of memory.	The p_elim == 13 overhead
    takes about 3 seconds and requires ~1.5 Megs of memory.

    The value p_elim == 17 is best for long factorizations.  It is the
    fastest even thought the initial startup overhead is larger than
    for p_elim == 13.


mod.cal

    lmod(a)
    mod_print(a)
    mod_one()
    mod_cmp(a, b)
    mod_rel(a, b)
    mod_add(a, b)
    mod_sub(a, b)
    mod_neg(a)
    mod_mul(a, b)
    mod_square(a)
    mod_inc(a)
    mod_dec(a)
    mod_inv(a)
    mod_div(a, b)
    mod_pow(a, b)

    Routines to handle numbers modulo a specified number.


natnumset.cal

    isset(a)
    setbound(n)
    empty()
    full()
    isin(a, b)
    addmember(a, n)
    rmmember(a, n)
    set()
    mkset(s)
    primes(a, b)
    set_max(a)
    set_min(a)
    set_not(a)
    set_cmp(a, b)
    set_rel(a, b)
    set_or(a, b)
    set_and(a, b)
    set_comp(a)
    set_setminus(a, b)
    set_diff(a,b)
    set_content(a)
    set_add(a, b)
    set_sub(a, b)
    set_mul(a, b)
    set_square(a)
    set_pow(a, n)
    set_sum(a)
    set_plus(a)
    interval(a, b)
    isinterval(a)
    set_mod(a, b)
    randset(n, a, b)
    polyvals(L, A)
    polyvals2(L, A, B)
    set_print(a)

    Demonstration of how the string operators and functions may be used
    for defining and working with sets of natural numbers not exceeding a
    user-specified bound.


pell.cal

    pellx(D)
    pell(D)

    Solve Pell's equation; Returns the solution X to: X^2 - D * Y^2 = 1.
    Type the solution to Pell's equation for a particular D.


pi.cal

    qpi(epsilon)
    piforever()

    The qpi() calculate pi within the specified epsilon using the quartic
    convergence iteration.

    The piforever() prints digits of pi, nicely formatted, for as long
    as your free memory space and system up time allows.

    The piforever() function (written by Klaus Alexander Seistrup
    <klaus@seistrup.dk>) was inspired by an algorithm conceived by
    Lambert Meertens.  See also the ABC Programmer's Handbook, by Geurts,
    Meertens & Pemberton, published by Prentice-Hall (UK) Ltd., 1990.


pix.cal

    pi_of_x(x)

    Calculate the number of primes < x using A(n+1)=A(n-1)+A(n-2).  This
    is a SLOW painful method ... the builtin pix(x) is much faster.
    Still, this method is interesting.


pollard.cal

    pfactor(N, N, ai, af)

    Factor using Pollard's p-1 method.


poly.cal

    Calculate with polynomials of one variable.	 There are many functions.
    Read the documentation in the resource file.


prompt.cal

    adder()
    showvalues(str)

    Demonstration of some uses of prompt() and eval().


psqrt.cal

    psqrt(u, p)

    Calculate square roots modulo a prime


qtime.cal

    qtime(utc_hr_offset)

    Print the time as English sentence given the hours offset from UTC.


quat.cal

    quat(a, b, c, d)
    quat_print(a)
    quat_norm(a)
    quat_abs(a, e)
    quat_conj(a)
    quat_add(a, b)
    quat_sub(a, b)
    quat_inc(a)
    quat_dec(a)
    quat_neg(a)
    quat_mul(a, b)
    quat_div(a, b)
    quat_inv(a)
    quat_scale(a, b)
    quat_shift(a, b)

    Calculate using quaternions of the form: a + bi + cj + dk.	In these
    functions, quaternions are manipulated in the form: s + v, where
    s is a scalar and v is a vector of size 3.


randbitrun.cal

    randbitrun([run_cnt])

    Using randbit(1) to generate a sequence of random bits, determine if
    the number and length of identical bits runs match what is expected.
    By default, run_cnt is to test the next 65536 random values.

    This tests the a55 generator.


randmprime.cal

    randmprime(bits, seed [,dbg])

    Find a prime of the form h*2^n-1 >= 2^bits for some given x.  The
    initial search points for 'h' and 'n' are selected by a cryptographic
    pseudo-random number generator.  The optional argument, dbg, if set
    to 1, 2 or 3 turn on various debugging print statements.


randombitrun.cal

    randombitrun([run_cnt])

    Using randombit(1) to generate a sequence of random bits, determine if
    the number and length of identical bits runs match what is expected.
    By default, run_cnt is to test the next 65536 random values.

    This tests the Blum-Blum-Shub generator.


randomrun.cal

    randomrun([run_cnt])

    Perform the "G. Run test" (pp. 65-68) as found in Knuth's "Art of
    Computer Programming - 2nd edition", Volume 2, Section 3.3.2 on
    the builtin rand() function.  This function will generate run_cnt
    64 bit values.  By default, run_cnt is to test the next 65536
    random values.

    This tests the Blum-Blum-Shub generator.


randrun.cal

    randrun([run_cnt])

    Perform the "G. Run test" (pp. 65-68) as found in Knuth's "Art of
    Computer Programming - 2nd edition", Volume 2, Section 3.3.2 on
    the builtin rand() function.  This function will generate run_cnt
    64 bit values.  By default, run_cnt is to test the next 65536
    random values.

    This tests the a55 generator.

repeat.cal

    repeat(digit_set, repeat_count)

    Return the value of the digit_set repeated repeat_count times.
    Both digit_set and repeat_count must be integers > 0.

    For example repeat(423,5) returns the value 423423423423423,
    which is the digit_set 423 repeated 5 times.


regress.cal

    Test the correct execution of the calculator by reading this resource
    file.  Errors are reported with '****' messages, or worse. :-)


screen.cal

    up
    CUU	/* same as up */
    down = CUD
    CUD	/* same as down */
    forward
    CUF	/* same as forward */
    back = CUB
    CUB	/* same as back */
    save
    SCP	/* same as save */
    restore
    RCP	/* same as restore */
    cls
    home
    eraseline
    off
    bold
    faint
    italic
    blink
    rapidblink
    reverse
    concealed
    /* Lowercase indicates foreground, uppercase background */
    black
    red
    green
    yellow
    blue
    magenta
    cyan
    white
    Black
    Red
    Green
    Yellow
    Blue
    Magenta
    Cyan
    White

    Define ANSI control sequences providing (i.e., cursor movement,
    changing foreground or background color, etc.) for VT100 terminals
    and terminal window emulators (i.e., xterm, Apple OS/X Terminal,
    etc.) that support them.

    For example:

	read screen
	print green:"This is green. ":red:"This is red.":black


seedrandom.cal

    seedrandom(seed1, seed2, bitsize [,trials])

    Given:
	seed1 - a large random value (at least 10^20 and perhaps < 10^93)
	seed2 - a large random value (at least 10^20 and perhaps < 10^93)
	size - min Blum modulus as a power of 2 (at least 100, perhaps > 1024)
	trials - number of ptest() trials (default 25) (optional arg)

    Returns:
	the previous random state

    Seed the cryptographically strong Blum generator.  This functions allows
    one to use the raw srandom() without the burden of finding appropriate
    Blum primes for the modulus.


set8700.cal

    set8700_getA1() defined
    set8700_getA2() defined
    set8700_getvar() defined
    set8700_f(set8700_x) defined
    set8700_g(set8700_x) defined

    Declare globals and define functions needed by dotest() (see
    dotest.cal) to evaluate set8700.line a line at a time.


set8700.line

    A line-by-line evaluation file for dotest() (see dotest.cal).
    The set8700.cal file (and dotest.cal) should be read first.


smallfactors.cal

    smallfactors(x0)
    printsmallfactors(flist)

    Lists the prime factors of numbers smaller than 2^32. Try for example:
    printsmallfactors(smallfactors(10!)).


solve.cal

    solve(low, high, epsilon)

    Solve the equation f(x) = 0 to within the desired error value for x.
    The function 'f' must be defined outside of this routine, and the
    low and high values are guesses which must produce values with
    opposite signs.


specialfunctions.cal

    beta(a,b)

    Calculates the value of the beta function.  See:

	https://en.wikipedia.org/wiki/Beta_function
        http://mathworld.wolfram.com/BetaFunction.html
        http://dlmf.nist.gov/5.12

    for information on the beta function.


    betainc(a,b,z)

    Calculates the value of the regularized incomplete beta function.  See:

	https://en.wikipedia.org/wiki/Beta_function
        http://mathworld.wolfram.com/RegularizedBetaFunction.html
        http://dlmf.nist.gov/8.17

    for information on the regularized incomplete beta function.


    expoint(z)

    Calculates the value of the exponential integral Ei(z) function at z.
    See:

	http://en.wikipedia.org/wiki/Exponential_integral
        http://www.cs.utah.edu/~vpegorar/research/2011_JGT/

    for information on the exponential integral Ei(z) function.


    erf(z)

    Calculates the value of the error function at z.  See:

	http://en.wikipedia.org/wiki/Error_function

    for information on the error function function.


    erfc(z)

    Calculates the value of the complementary error function at z.  See:

	http://en.wikipedia.org/wiki/Error_function

    for information on the complementary error function function.


    erfi(z)

    Calculates the value of the imaginary error function at z.  See:

	http://en.wikipedia.org/wiki/Error_function

    for information on the imaginary error function function.


    erfinv(x)

    Calculates the inverse of the error function at x.  See:

	http://en.wikipedia.org/wiki/Error_function

    for information on the inverse of the error function function.


    faddeeva(z)

    Calculates the value of the complex error function at z.  See:

	http://en.wikipedia.org/wiki/Faddeeva_function

    for information on the complex error function function.


    gamma(z)

    Calculates the value of the Euler gamma function at z.  See:

	http://en.wikipedia.org/wiki/Gamma_function
        http://dlmf.nist.gov/5

    for information on the Euler gamma function.


    gammainc(a,z)

    Calculates the value of the lower incomplete gamma function for
    arbitrary a, z.  See:

	http://en.wikipedia.org/wiki/Incomplete_gamma_function

    for information on the lower incomplete gamma function.

    gammap(a,z)

    Calculates the value of the regularized lower incomplete gamma
    function for a, z with a not in -N.  See:

	http://en.wikipedia.org/wiki/Incomplete_gamma_function

    for information on the regularized lower incomplete gamma function.

    gammaq(a,z)

    Calculates the value of the regularized upper incomplete gamma
    function for a, z with a not in -N.  See:

	http://en.wikipedia.org/wiki/Incomplete_gamma_function

    for information on the regularized upper incomplete gamma function.


    heavisidestep(x)

    Computes the Heaviside stepp function (1+sign(x))/2


    harmonic(limit)

    Calculates partial values of the harmonic series up to limit.  See:

	http://en.wikipedia.org/wiki/Harmonic_series_(mathematics)
        http://mathworld.wolfram.com/HarmonicSeries.html

    for information on the harmonic series.


    lnbeta(a,b)

    Calculates the natural logarithm of the beta function.  See:

	https://en.wikipedia.org/wiki/Beta_function
        http://mathworld.wolfram.com/BetaFunction.html
        http://dlmf.nist.gov/5.12

    for information on the beta function.

    lngamma(z)

    Calculates the value of the logarithm of the Euler gamma function
    at z.  See:

	http://en.wikipedia.org/wiki/Gamma_function
        http://dlmf.nist.gov/5.15

    for information on the derivatives of the the Euler gamma function.


    polygamma(m,z)

    Calculates the value of the m-th derivative of the Euler gamma
    function at z.  See:

	http://en.wikipedia.org/wiki/Polygamma
        http://dlmf.nist.gov/5

    for information on the n-th derivative ofthe Euler gamma function. This
    function depends on the script zeta2.cal.


    psi(z)

    Calculates the value of the first derivative of the Euler gamma
    function at z.  See:

	http://en.wikipedia.org/wiki/Digamma_function
        http://dlmf.nist.gov/5

    for information on the first derivative of the Euler gamma function.


    zeta(s)

    Calculates the value of the Rieman Zeta function at s.  See:

	http://en.wikipedia.org/wiki/Riemann_zeta_function
        http://dlmf.nist.gov/25.2

    for information on the Riemann zeta function. This function depends
    on the script zeta2.cal.


statistics.cal

    gammaincoctave(z,a)

    Computes the regularized incomplete gamma function in a way to
    correspond with the function in Octave.

    invbetainc(x,a,b)

    Computes the inverse of the regularized beta function. Does so the
    brute-force way wich makes it a bit slower.

    betapdf(x,a,b)
    betacdf(x,a,b)
    betacdfinv(x,a,b)
    betamedian(a,b)
    betamode(a,b)
    betavariance(a,b)
    betalnvariance(a,b)
    betaskewness(a,b)
    betakurtosis(a,b)
    betaentropy(a,b)
    normalpdf(x,mu,sigma)
    normalcdf(x,mu,sigma)
    probit(p)
    normalcdfinv(p,mu,sigma)
    normalmean(mu,sigma)
    normalmedian(mu,sigma)
    normalmode(mu,sigma)
    normalvariance(mu,sigma)
    normalskewness(mu,sigma)
    normalkurtosis(mu,sigma)
    normalentropy(mu,sigma)
    normalmgf(mu,sigma,t)
    normalcf(mu,sigma,t)
    chisquaredpdf(x,k)
    chisquaredpcdf(x,k)
    chisquaredmean(x,k)
    chisquaredmedian(x,k)
    chisquaredmode(x,k)
    chisquaredvariance(x,k)
    chisquaredskewness(x,k)
    chisquaredkurtosis(x,k)
    chisquaredentropy(x,k)
    chisquaredmfg(k,t)
    chisquaredcf(k,t)

    Calculates a bunch of (hopefully) aptly named statistical functions.


strings.cal

    isascii(c)
    isblank(c)

    Implements some of the functions of libc's ctype.h and strings.h.

    NOTE: A number of the ctype.h and strings.h functions are now builtin
          functions in calc.

   WARNING: If the remaining functions in this calc resource file become
	    calc builtin functions, then strings.cal may be removed in
	    a future release.


sumsq.cal

    ss(p)

    Determine the unique two positive integers whose squares sum to the
    specified prime.  This is always possible for all primes of the form
    4N+1, and always impossible for primes of the form 4N-1.


sumtimes.cal

    timematsum(N)
    timelistsum(N)
    timematsort(N)
    timelistsort(N)
    timematreverse(N)
    timelistreverse(N)
    timematssq(N)
    timelistssq(N)
    timehmean(N,M)
    doalltimes(N)

    Give the user CPU time for various ways of evaluating sums, sums of
    squares, etc, for large lists and matrices.  N is the size of
    the list or matrix to use.  The doalltimes() function will run
    all fo the sumtimes tests.  For example:

    	doalltimes(1e6);


surd.cal

    surd(a, b)
    surd_print(a)
    surd_conj(a)
    surd_norm(a)
    surd_value(a, xepsilon)
    surd_add(a, b)
    surd_sub(a, b)
    surd_inc(a)
    surd_dec(a)
    surd_neg(a)
    surd_mul(a, b)
    surd_square(a)
    surd_scale(a, b)
    surd_shift(a, b)
    surd_div(a, b)
    surd_inv(a)
    surd_sgn(a)
    surd_cmp(a, b)
    surd_rel(a, b)

    Calculate using quadratic surds of the form: a + b * sqrt(D).


test1700.cal

    value

    This resource files is used by regress.cal to test the read and
    use keywords.


test2600.cal

    global defaultverbose
    global err
    testismult(str, n, verbose)
    testsqrt(str, n, eps, verbose)
    testexp(str, n, eps, verbose)
    testln(str, n, eps, verbose)
    testpower(str, n, b, eps, verbose)
    testgcd(str, n, verbose)
    cpow(x, n, eps)
    cexp(x, eps)
    cln(x, eps)
    mkreal()
    mkcomplex()
    mkbigreal()
    mksmallreal()
    testappr(str, n, verbose)
    checkappr(x, y, z, verbose)
    checkresult(x, y, z, a)
    test2600(verbose, tnum)

    This resource files is used by regress.cal to test some of builtin
    functions in terms of accuracy and roundoff.


test2700.cal

    global defaultverbose
    mknonnegreal()
    mkposreal()
    mkreal_2700()
    mknonzeroreal()
    mkposfrac()
    mkfrac()
    mksquarereal()
    mknonsquarereal()
    mkcomplex_2700()
    testcsqrt(str, n, verbose)
    checksqrt(x, y, z, v)
    checkavrem(A, B, X, eps)
    checkrounding(s, n, t, u, z)
    iscomsq(x)
    test2700(verbose, tnum)

    This resource files is used by regress.cal to test sqrt() for real and
    complex values.


test3100.cal

    obj res
    global md
    res_test(a)
    res_sub(a, b)
    res_mul(a, b)
    res_neg(a)
    res_inv(a)
    res(x)

    This resource file is used by regress.cal to test determinants of
    a matrix.


test3300.cal

    global defaultverbose
    global err
    testi(str, n, N, verbose)
    testr(str, n, N, verbose)
    test3300(verbose, tnum)

    This resource file is used by regress.cal to provide for more
    determinant tests.


test3400.cal

    global defaultverbose
    global err
    test1(str, n, eps, verbose)
    test2(str, n, eps, verbose)
    test3(str, n, eps, verbose)
    test4(str, n, eps, verbose)
    test5(str, n, eps, verbose)
    test6(str, n, eps, verbose)
    test3400(verbose, tnum)

    This resource file is used by regress.cal to test trig functions.
    containing objects.

test3500.cal

    global defaultverbose
    global err
    testfrem(x, y, verbose)
    testgcdrem(x, y, verbose)
    testf(str, n, verbose)
    testg(str, n, verbose)
    testh(str, n, N, verbose)
    test3500(verbose, n, N)

    This resource file is used by regress.cal to test the functions frem,
    fcnt, gcdrem.


test4000.cal

    global defaultverbose
    global err
    global BASEB
    global BASE
    global COUNT
    global SKIP
    global RESIDUE
    global MODULUS
    global K1
    global H1
    global K2
    global H2
    global K3
    global H3
    plen(N) defined
    rlen(N) defined
    clen(N) defined
    ptimes(str, N, n, count, skip, verbose) defined
    ctimes(str, N, n, count, skip, verbose) defined
    crtimes(str, a, b, n, count, skip, verbose) defined
    ntimes(str, N, n, count, skip, residue, mod, verbose) defined
    testnextcand(str, N, n, cnt, skip, res, mod, verbose) defined
    testnext1(x, y, count, skip, residue, modulus) defined
    testprevcand(str, N, n, cnt, skip, res, mod, verbose) defined
    testprev1(x, y, count, skip, residue, modulus) defined
    test4000(verbose, tnum) defined

    This resource file is used by regress.cal to test ptest, nextcand and
    prevcand builtins.


test4100.cal

    global defaultverbose
    global err
    global K1
    global K2
    global BASEB
    global BASE
    rlen_4100(N) defined
    olen(N) defined
    test1(x, y, m, k, z1, z2) defined
    testall(str, n, N, M, verbose) defined
    times(str, N, n, verbose) defined
    powtimes(str, N1, N2, n, verbose) defined
    inittimes(str, N, n, verbose) defined
    test4100(verbose, tnum) defined

    This resource file is used by regress.cal to test REDC operations.


test4600.cal

    stest(str [, verbose]) defined
    ttest([m, [n [,verbose]]]) defined
    sprint(x) defined
    findline(f,s) defined
    findlineold(f,s) defined
    test4600(verbose, tnum) defined

    This resource file is used by regress.cal to test searching in files.


test5100.cal

    global a5100
    global b5100
    test5100(x) defined

    This resource file is used by regress.cal to test the new code generator
    declaration scope and order.


test5200.cal

    global a5200
    static a5200
    f5200(x) defined
    g5200(x) defined
    h5200(x) defined

    This resource file is used by regress.cal to test the fix of a
    global/static bug.


test8400.cal

    test8400() defined

    This resource file is used by regress.cal to check for quit-based
    memory leaks.


test8500.cal

    global err_8500
    global L_8500
    global ver_8500
    global old_seed_8500
    global cfg_8500
    onetest_8500(a,b,rnd) defined
    divmod_8500(N, M1, M2, testnum) defined

    This resource file is used by regress.cal to the // and % operators.


test8600.cal

    global min_8600
    global max_8600
    global hash_8600
    global hmean_8600

    This resource file is used by regress.cal to test a change of
    allowing up to 1024 args to be passed to a builtin function.


test8900.cal

    This function tests a number of calc resource functions contributed
    by Christoph Zurnieden.  These include:

	bernpoly.cal
	brentsolve.cal
	constants.cal
	factorial2.cal
	factorial.cal
	lambertw.cal
	lnseries.cal
	specialfunctions.cal
	statistics.cal
	toomcook.cal
	zeta2.cal


unitfrac.cal

    unitfrac(x)

    Represent a fraction as sum of distinct unit fractions.


toomcook.cal


    toomcook3(a,b)
    toomcook4(a,b)

    Toom-Cook multiplication algorithm.  Multiply two integers a,b by
    way of the Toom-Cook algorithm.  See:

	http://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication

    toomcook3square(a)
    toomcook4square(a)

    Square the integer a by way of the Toom-Cook algorithm.  See:

	http://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication

    The function toomCook4(a,b) calls the function toomCook3(a,b) which
    calls built-in multiplication at a specific cut-off point. The
    squaring functions act in the same way.


varargs.cal

    sc(a, b, ...)

    Example program to use 'varargs'.  Program to sum the cubes of all
    the specified numbers.


xx_print.cal

    is_octet(a) defined
    list_print(a) defined
    mat_print (a) defined
    octet_print(a) defined
    blk_print(a) defined
    nblk_print (a) defined
    strchar(a) defined
    file_print(a) defined
    error_print(a) defined

    Demo for the xx_print object routines.


zeta2.cal

    hurwitzzeta(s,a)

    Calculate the value of the Hurwitz Zeta function.  See:

	http://en.wikipedia.org/wiki/Hurwitz_zeta_function
        http://dlmf.nist.gov/25.11

    for information on this special zeta function.


## Copyright (C) 2000,2014,2017  David I. Bell and Landon Curt Noll
##
## Primary author: Landon Curt Noll
##
## Calc is open software; you can redistribute it and/or modify it under
## the terms of the version 2.1 of the GNU Lesser General Public License
## as published by the Free Software Foundation.
##
## Calc is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU Lesser General
## Public License for more details.
##
## A copy of version 2.1 of the GNU Lesser General Public License is
## distributed with calc under the filename COPYING-LGPL.  You should have
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
##
## Under source code control:	1990/02/15 01:50:32
## File existed as early as:	before 1990
##
## chongo <was here> /\oo/\	http://www.isthe.com/chongo/
## Share and enjoy!  :-)	http://www.isthe.com/chongo/tech/comp/calc/