 Do
 I prefer and use these functions over the official Scheme syntactic sugar.
(((fileVal "Do") 'Do) 6 (lambda (j) (write (+ j 2)))) ; => 765432
(((fileVal "Do") 'DoL) 5 (lambda (j) (* j 2))) ; => (0 2 4 6 8)
(((fileVal "Do") 'DoV) 4 (lambda (j) (+ j 4))) ; => #(4 5 6 7)
More info
 Equal
 An equivalence test for potentially infinite Scheme structures.
More info
 RC4
 A source of quality pseudo random generators based on the RC4 crypto algorithm.
((fileVal "RC4") seedstring) returns a mutable object whose initial state is determined by the string argument.
The state is the abstracted RC4 state.
(let ((g (((fileVal "RC4") "Seed stuff") 'nb))) (list (g 4) (g 3) (g 2)))
yields (3453575773 3264371 6246).
(g n) yields integers uniformly distributed in [0, 2^{8n}).
(let ((g (((fileVal "RC4") "Seed stuff") 'sb))) (list (g) (g) (g)))
yields (205 217 98), integers uniformly distributed in [0, 2^{8}).
These successive values are the eight bit bytes of the RC4 keystream.
((((fileVal "RC4") "Seed stuff") 'rbi) n)
for n>1 yields a generator of integers uniformly distributed in [0, n).
(let* ((A ((fileVal "RC4") "Seed")) (B (A 'child))
(C (B 'sb))) (((fileVal "Do") 'DoL) 8 (lambda(j) (C))))
=> (135 220 170 236 64 49 35 83)
B is a child of A and behaves just as A.
It carries its own mutable state which is seeded from A’s state which is thereby modified.
More info
 AGEC and AGECp
 Abelian Group from Elliptic Curve; See this.
 RND
 Generator of sequences of random normal deviates.
(let ((sg ((fileVal "RND") "frSeed"))) (let r ((n 10000) (m 0) (s 0))
(if (zero? n) (cons m s) (let ((w (sg))) (r ( n 1) (+ m w) (+ s (* w w)))))))
yields (41.62913630233171 . 10215.783961421654).
 rr
 Random rational generator
(let ((g ((fileVal "rr") "Seed"))) (list (g) (g)))
yields (37/113 125/23).
 binomial
 Binomial coefficients: ((fileVal "binomial") p q) = C(p, q).
 egcd
 Extended Euclidean Algorithm.
((fileVal "egcd") a b) yields (x . y) such that ax+by = (gcd a b)
((fileVal "egcd") 100 144) => (13 . 9)
gcd(100, 144) = 4, 1300 − 1296 = 4
 Factorial
 ((fileVal "Factorial") n) yields n! for n > −1.
It works for integral and half integral values.
((fileVal "Factorial") 1/2) => (−1/2)! = 1.7724538509055159 = √π.
 expt
 ((fileVal "expt") op one inv) returns a function pow such that (pow k n) returns k^{n} where n is an integer, k is in some algebra, op is an associative operator in that algebra, and the following rules hold:
(pow e 0) = one
(pow e 1) = e
(pow e (+ m n)) = (op (pow e m) (pow e n))
(eq? (op e (inv e)) one)
The provided inv function will be invoked only for negative powers.
The time is the product of an op time and the log of n.
 gamma

Definition:  
Calculation:  
The incomplete Gamma function
((fileVal "gamma") s z) => γ(s, z).
For constant s and large values of z γ(s, z) is asymptotic to some constant.
Power series don’t do this well.
This Algol68 code can run in multiple precision so as to extend the range.
 Chi2
 The χ^{2} distribution, both density:
(((fileVal "Chi2") 'f) x k) = f(x; k)
and cumulative:
(((fileVal "Chi2") 'F) x k) = F(x; k)
A meagre corroboration
 int
 ((fileVal "int") x0 x1 k f) yields a crude integral of f using k equally spaced points from x0 to x1.
 pias
 To search an arithmetic sequence for the first prime.
Candidates are tested with the Miller−Rabin primality test.
((fileVal "pias") a b)
returns the first prime among a + nb for n = 0, 1, 2, ... .
The value ("Always divisible by" n) is returned instead if a and b are both divisible by n and n > 1.
If a and b are relatively prime there are such primes.
There is extraneous printed output during the computation consisting of a comma for each candidate which is excluded for being divisible by a prime less than 18, and a construct such as (123 #f) when the Miller−Rabin test finds the candidate to be composite for n = 123.
Running this program for largish numbers suggests that the probability of false positives is very much less than 1/2 which is the proven upper bound on the probability.
Indeed I have seen no false positives in a few hundred cases.
((fileVal "pias") (expt 10 100) 1)
indicates that 10^{100} − 797 is the largest prime less than 10^{100}.
(10^{1000}−1769 is prime too.)
more info.
 piasd
 like pias above but if result is p then both p and 2p+1 are prime.
 modexp
 ((fileVal "modexp") p q m) yields p^{q} modulo m.
p, q and m are integers: q ≥ 0 and m ≥ 2.
 Miller−Rabin
 The Miller−Rabin primality test:
If p is prime and 1 < a < p−1 then (((fileVal "MillerRabin") 'base) a p) is true.
If p is composite then it is false for almost all such a’s.
Cumulative count of false positives less than 5^{n} for the composite 2^{25}+1: (3 5 7 10 12 14 19 25 … )
((((fileVal "MillerRabin") 'multi) n) p) chooses n random probes to test p for primality.
#t is returned if each MillerRabine probe reports that p is prime.
(((fileVal "MillerRabin") 'multi) n) denotes the object that is mutable by virtue of carrying the PRNG to choose the random probes.
 try
 Conventional trycatch function.
((fileVal "try") (lambda (e) (write "working") 'w)
(lambda (t) (write (list "exception" t)) 'e))
; => "working"w
((fileVal "try") (lambda (e) (write "attempting") (e "peculiar") (write "don’t get here!"))
(lambda (t) (write (list "exception" t)) 'e))
; => "attempting"("exception" "peculiar")e
e above is a continuation.
See More info.
 gr
 Changes a function of zots into a function of lists of zots, returning a list of results.
If gr = (fileVal "gr") then ((gf odd?) '(1 2 3 4 5)) yields (#t #f #t #f #t) and ((gr +) '(2 3 4 5) '(9 7 5 3)) yields (11 10 9 8).
 transpose
 (fileVal "transpose") transposes a matrix expressed as a nonempty list of equal length lists.
 transpose0
 ((fileVal "transpose0") 0 zero?) is a function that transposes an infinite list of infinite lists where all but a finite number of terms are terminal zeros and omitted.
Trailing all zero rows are also omitted.
Adding a 0 to the end of a matrix row or adding an empty row to the end of a matrix results in a Scheme value denotation the same infinite matrix.
More info
 sort

((fileVal "sort") '(5 3 4 882 3 5 44 3 9 7) <) => (7 3 3 3 4 5 5 9 44 882)
 Set
 A suite of set functions.
Transcribed from a source with a GNU license.
The suite is parameterized by a total ordering function for set elements.
If sy is a symbol from the list of values listed here, then (((fileVal "Set") comp) 'sy) yields the value described on the same page.
(comp a b) must return a negative number, 0, or a positive number depending on whether a<b, a=b or a>b respectively.
((fileVal "Set") ) returns a map from symbols to set tools, in this case sets of numbers.
If m is such a map then (m 'union) returns the setunion operator.
See this for more detail.
 Farey
 Farey Sequence: ((fileVal "Farey") n) => F_{n}.
 ContFrac
 Continued Fractions:
If x is (fileVal "ContFrac") and n is a positive integer and y is a positive number then ((car x) y n) produces a list of up to n terms of the continued fraction for y.
The list may be shorter if x is an exact rational.
If z>0 and n is large enough then ((cdr x) ((car x) z n)) should yield z within floating point precision.
 stream
 Routines to convert between streams and lists.
(ylppa (fileVal "stream") (lambda (sm si SG>lst lst>SG fs Cart string>SG)
; Scope of stream functions
(SG>lst (sm (si 6) )) ; => (0 1 2 3 4 5)
))
more detail
 BiasBin
 Conversion to and from biased radix 2 representation.
If 0<b<1, 0<x<1 and n is a nonnegative integer then
(let ((p ((fileVal "BiasBin") b))) (= ((cdr p) ((car p) x n)) x))
yields true, especially if b and x are exact.
more info
 Matrix
 Determinants and matrix functions over arbitrary fields.
A matrix is a list of vectors and vector is a list of field elements.
Arguments to (fileVal "Matrix")
 Random scalar generator
 the zero element for the field
 a predicate that tests its field argument for being zero
 the multiplicative identity of the field
 the operator that adds two field arguments
 the unary additive inverse function
 the operator that multiplies two field arguments
 the multiplicative inverse function
Yields the list:
 Random n×n Matrix Generator (rmg n)
 Matrix multiplication
 Matrix inversion (2nd arg is continuation for singular matrix)
 Inner product of two vectors
 Transpose
 Determinant
 Identity test
 Binary vector equality test
 Binary matrix equality test
(rmg k) returns a random k by k matrix.
They may deployed:
(ylppa ((fileVal "Matrix") '() 0 zero? 1 +  * /)
(lambda (rm matm matinv ip tr det i? v= m=)
; Scope of matrix functions over Scheme numbers
(det '((1 3 2)(4 0.6 2)(3 8 4)))
)) ; => 16.8
If the ‘Matrix Generator’ is not used then the ‘Random scalar generator’ is not used.
Note that we do not supply a good random scalar generator since we do not need a random matrix generator.
The 2nd argument to the produced inversion routine is a continuation which is invoked for a singular matrix, passing a vector which, when padded with sufficient zeros, yields x such that Mx = 0.
See this for critical application of this value.
(ylppa ((fileVal "Matrix") '() 0 zero? 1 +  * /)
(lambda (rm matm matinv ip tr det i? v= m=)
((fileVal "Try") (lambda (e) (matinv '((2 3 5) (6 9 4) (8 12 9)) e))
(lambda (v) (write v)))))
yields (3/2 1) since the first two columns are dependent.
The code inverts matrices of quaternions; see quatApp below.
Thus it probably handles division rings.
It even inverts matrices in associative algebras with dense inverses such as Clifford Matrices.
More info
 MatInverse
 A concrete matrix invert built from above.
((fileVal "MatInverse") '((2 4 5)(3 2 5)(5 7 4)))
yields ((27/53 19/53 10/53) (13/53 17/53 5/53) (11/53 6/53 8/53)).
Any Scheme numbers may be used including, rational, complex and floating point.
 GenDiag
 ((fileVal "GenDiag") n d b) returns a list of n lists each with n elements.
The jth elements of the jth list is d and the other elements are b.
((fileVal "GenDiag") n 1 0) returns the n by n identity matrix.
 HilbertMat & HilbertM
 ((fileVal "HilbertMat") n) produces the infamous nth Hilbert matrix and
((fileVal "HilbertM") n) inverts that and reports its determinant and inverse.
 diag
 ((fileVal "diag") numlist) yields a diagonal matrix with elements of numlist down the diagonal.
 ranoth
 ((fileVal "ranoth") "ran seed" n) yields a generator of random orthogonal n by n real matrices, i.e. matrices from O(n).
The distribution is invariant over O(n).
Adapted from this.
Notes on testing.
 gRREF
 Reduced Row Echelon Form of matrix parameterized by field.
(let ((R ((fileVal "gRREF") zero? 0 1 / *  C)))
((R 'rref) m) ; Reduced Row Echelon Form of matrix m.
((R 'rank) m) ; rank of m
((R 'minspan) m) ; spanning subset of m
)
Arguments  and * are binary; / is unary.
Argument C is for ‘canonical’.
If C is true then the resulting transformation omits trailing zero vectors and this modification makes the matrix canonical with respect to which subspace its vectors span.
The classical definition of RREF requires as many rows in the output as in the input.
The returned RREF function returns the classical answer if C is false (#f).
more info
 Intersect
 Compute intersection of subspaces of vector space where subspaces are denoted by a lists of spanning vectors.
((fileVal "Intersect") a b) yields the intersection.
See demos at end of source and here.
Detailed info
 gIntersect
 Specification.
Compute intersection of subspaces of vector space over field where subspaces are identified by a list of spanning vectors.
See demos at end of source and here.
If A is the matrix of the coefficients of a set of homogeneous linear equations then ((((fileVal "gIntersect") 0 zero? 1 +  * /) 'oss) A 0) yields a minimal list of solutions that span the space of all solutions.
more info
 testInter
 Code to test gIntersect above.
More info.
 PGc
 Projective Geometry: Specs
 PGLat
 Verifying lattice properties of subspaces
 Desargues
 Corroborating Desargues’ theorem.
comments
 Polynomial tools
 Several tools where given a ring R, a polynomial ring K[R] is returned and given a field a gcd is provided.
A transcendental field extension is also provided.
(fileVal "transpose0") will be useful in some related applications.
Some terminology and fundamental notions.
 finFieldMat
 ((fileVal "finFieldMat") p q n) yields some tools for n by n matrices whose elements are from GF(p^{q}).
The yield is (random mp inv ip tr deter it? v= m=).
Returned values are those of Matrix.
 finFieldMatTest
 ((fileVal "finFieldMatTest") p q n) runs a test of the above finite field n by n matrix logic from GF(p^{q}).
 Jacobi
 ((fileVal "Jacobi") a n) is the Jacobi symbol sometimes written .
 DivisionAlgebra
 (fileVal "DivisionAlgebra") automates the CayleyDiskson construction of the Division Algebras.
(fileVal "DivisionAlgebra") yields the list (G reals e) where
(G(G reals)) yields the Hamiltonians, for example and (e 4 j) yields e_{j} which is the jth basis element of the level four algebra.
More info
 k4ZerDiv
 Zero Divisors in Sedenions
 divAlgSurvey
 (fileVal "divAlgSurvey") builds the first five CayleyDiskson algebras and reports their algebraic properites.
More info
 quaternion
 (fileVal "quaternion") yields a list of quaternion tools:
(random quaternion generator, zero, zero predicate, one, addition, negation, multiplication, inverse).
Here is an alternate abinitio construction of the quaternions.
 quatApp
 (fileVal "quatApp") builds matrices of quaternions and successfully inverts them with Matrix!
 GFp
 For a prime p, ((fileVal "GFp") p) yields a list of the four operations over GF(p):
+, −, *, /.
The ops − and / are unary, the operands are non negative integers less than p.
 GFsqrt
 is a modular square root routine.
If p is prime then (((fileVal "GFsqrt") p) (* r r)) yields r or p−r.
If x is not a quadratic residue modulo p then it returns 'none.
This addresses the other finite fields.
 finiteField
 Code to perform arithmetic over any finite field.
This module exports tools of interest to builders of alternate representations.
Detailed specs are here.
Most users want GFpq below.
 GFpq
 ((fileVal "GFpq") p q st)
returns a tool list
 sample generator, (() → F)
 zero, (F)
 zero predicate, (F → bool)
 one, (F)
 add, (F×F → F)
 negate, (F → F)
 times, (F×F → F)
 invert (F → F)
such as needed for input to the Matrix module.
The finite field is GF(p^{q}) and st is a nonempty string which only seeds the random ‘sample generator’.
The following illustrates matrices from Finite fields:
(ylppa (ylppa ((fileVal "GFpq") 17 4 "p") (fileVal "Matrix"))
(lambda (rmg mm inv ip tr det i? v= m=)
(let ((inv (lambda (q) ((fileVal "try") (lambda (w) (inv q w))(lambda (e) "e")))))
(let* ((m (rmg 4))(mi (inv m))) (list m mi (inv mi) (i? (mm m mi)))))))
=> (…, …, …, #t)
 rip
 ((fileVal "rip") p q) searches randomly for irreducible polynomials of degree q over GF(p).
Perhaps there are no 5th order irreducible binomials over GF(6700417) and searching lexicographically is thus frustrating.
They are abundant further on.
((fileVal "rip") 6700417 5) yields (8 . #6(5538386 4064128 2588321 4310749 3668254 1)).
It took 8 trials to find this irreducible polynomial.
 ffqr
 ((fileVal "ffqr") p q) counts quadratic residues in finite field GF(p^{q}).
See theorem.
 LagCoef
 ((fileVal "LagCoef") '() 0 zero? 1 +  * /) provides a function LC over the field of Scheme numbers that produces Lagrange coefficients.
Like "Matrix", "LagCoef" can take other field tools as illustrated in the last demo.
 Clifford0
 Simplest Clifford algebra. => (functor reals) where functor generates C^{n+1} from C^{n} and reals is the reals.
More info
 CliffTst0
 Simple corroboration of above.
More info
 Clifford1
 Fuller Clifford algebra.
More info
 CliffTst1
 Simple corroboration of above.
More info
 Clifford2
 Fuller Clifford algebra with pretty good division.
More info
 CliffTst2
 Simple corroboration of above.
More info
 IndCliff
 Clifford studied algebras based on negative definite quadratic forms.
This package generates ‘Clifford algebras’ based on indefinite forms.
((fileVal "IndCliff") (list +   )) generates a pseudo
Clifford algebra where γ_{0}^{2} = 1 instead of −1.
See test code at end of file for the returned values.
 IndCliffTst
 Demonstration of isometry with explanation.
 CliffMat
 (fileVal "CliffMat") Generates a 3×3 matrix of Clifford numbers (Cl(3)) using small rationals.
It inverts the matrix using the ‘almost everywhere inverse’.
It checks that the inverse times the original is the identity.
 CliffMat2
 (fileVal "CliffMat2") Generates a 5×5 matrix of Clifford numbers (Cl(4)) using random normal deviates as constituent floating point reals.
It inverts the matrix using the ‘almost everywhere inverse’.
It checks that the inverse times the original is the identity within a tolerance of 2^{−48}.
 Shannon
 Shannon’s information formula for a symbol chosen from a given probability distribution.
Argument is list of probabilities which should sum to 1.
 SRFI60
 A standard but not required extension to Scheme that exposes the bits in integers.
(fileVal "SRFI60") returns a list of the 19 functions specified in the SRFI60.
If xxx is a body of code using the preferred names in the SRFI then (apply (lambda (logand logior logxor lognot bitwiseif logtest logcount
integerlength log2binaryfactors logbit? copybit bitfield
copybitfield ash rotatebitfield reversebitfield integer>list
list>integer booleans>integer) xxx) (fileVal "SRFI60")) performs xxx with those 19 bindings.
This exploits the partial implementation of MzScheme.