jacal

— System: i386 GNU/Linux with Redhat Package Manager (rpm)

          wget http://swiss.csail.mit.edu/ftpdir/scm/scm-5e5-1.i386.rpm
          wget http://swiss.csail.mit.edu/ftpdir/scm/slib-3b1-1.noarch.rpm
          wget http://swiss.csail.mit.edu/ftpdir/scm/jacal-1b9-1.noarch.rpm
          rpm -U scm-5e5-1.i386.rpm slib-3b1-1.noarch.rpm jacal-1b9-1.noarch.rpm
          rm scm-5e5-1.i386.rpm slib-3b1-1.noarch.rpm jacal-1b9-1.noarch.rpm
     

The command `jacal' will start an interactive session.

— System: Unix
— System: GNU/Linux

          wget http://swiss.csail.mit.edu/ftpdir/scm/scm-5e5.zip
          wget http://swiss.csail.mit.edu/ftpdir/scm/slib-3b1.zip
          wget http://swiss.csail.mit.edu/ftpdir/scm/jacal-1b9.zip
          unzip -ao scm-5e5.zip
          unzip -ao slib-3b1.zip
          unzip -ao jacal-1b9.zip
          (cd slib; make install)
          (cd scm; make scm; make install)
          (cd jacal; make install)
          rm scm-5e5.zip slib-3b1.zip jacal-1b9.zip
     

The command `jacal' will start an interactive session using ELK, Gambit, Guile, Larceny, MIT-Scheme, MzScheme, Scheme48, SCM, or SISC. Type `jacal --help' for instructions.

— System: Apple

http://www.io.com/~cobblers/scm/ has downloads and utilities for installing SCM and SLIB on Macintosh computers.

— System: x86 Microsoft

Download and run http://swiss.csail.mit.edu/ftpdir/scm/SLIB-3b1-1.exe,
http://swiss.csail.mit.edu/ftpdir/scm/SCM-5e5-1.exe, and
http://swiss.csail.mit.edu/ftpdir/scm/JACAL-1b9-1.exe.

     (slib:load "/usr/local/lib/jacal/math")

     JACAL version 1b9, Copyright 1989-1999, 2002 Aubrey Jaffer
     JACAL comes with ABSOLUTELY NO WARRANTY; for details type `(terms)'.
     This is free software, and you are welcome to redistribute it
     under certain conditions; type `(terms)' for details.
     ;;; Type (math) to begin.

     (math)
     =>
     type qed; to return to scheme, type help; for help.
     e0 :

     Copyright © 2000,2001,2002 Free Software Foundation, Inc.
     51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA
     
     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

       Copyright (C)  year  your name.
       Permission is granted to copy, distribute and/or modify this document
       under the terms of the GNU Free Documentation License, Version 1.2
       or any later version published by the Free Software Foundation;
       with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
       Texts.  A copy of the license is included in the section entitled ``GNU
       Free Documentation License''.

         with the Invariant Sections being list their titles, with
         the Front-Cover Texts being list, and with the Back-Cover Texts
         being list.

— Operator: + augend addend

Addition of scalar quantities or componentwise addition of bunches is accomplished by means of the infix operator +. For example,

          e2 : a:[[1, 3, 5], [2, 4, 7]];
          
              [1  3  5]
          e2: [       ]
              [2  4  7]
          
          e3 : b:[2, 4];
          
          e3: [2, 4]
          
          e4 : a + b;
          
              [3  5  7 ]
          e4: [        ]
              [6  8  11]
          
          e5 : 3 + 2;
          
          e5: 5
          
          e6 : c + b;
          
          e6: [2 + c, 4 + c]
          
          e7 : e1 + e5;
          
                   2   2
          e7: 5 + (8 a + 12 a ) b
     

— Operator: - minuend subtrahend
— Operator: - subtrahend

The symbol - is used to denote either the binary infix operator subtraction or the unary minus.

          e1 : -[1,2,3];
          
          e1: [-1, -2, -3]
          
          e2 : 3-7;
          
          e2: -4
     

— Operator: +/- minuend subtrahend
— Operator: -/+ minuend subtrahend
— Operator: +/- augend
— Operator: -/+ augend

Jacal allows the use of +/- and -/+ as ambiguous signs (unary plus-or-minus, unary minus-or-plus) and as ambiguous infix operators (binary plus-or-minus, binary minus-or-plus). The value +/- is also represented by the constant %sqrt1, while -/+ is represented by -%sqrt1.

          e7 : u:+/-3;
          
          e7: 3 %sqrt1
          
          e8 : u^2;
          
          e8: 9
          
          e9 : +/-(u);
          
          e9: 3
          
          e10 : u-/+3;
          
          e10: b-/+(3 %sqrt1, 3)
     

— Operator: * multiplicand1 multiplicand2

Multiplication of scalar expressions such as numbers, polynomials, rational functions and algebraic functions is denoted by the infix operator *. For example,

          e1 : (2 + 3 * a) * 4 * a * b^2;
          
                         2   2
          e1: (8 a + 12 a ) b
     

One can also use * as an infix operator on bunches. In that case, it operates componentwise, in an appropriate sense. If the bunches are square matrices, the operator * multiplies corresponding entries of the two factors. It does not perform matrix multiplication. To multiply matrices one instead uses the operator . (i.e., a period). More generally, any binary scalar operator other than ^ can be used on bunches and acts componentwise.

— Operator: / dividend divisor

The symbol for division in Jacal is /. For example, the value returned by 6 / 2 is 3.

          e3 : (x^2 - y^2) / (x - y);
          
          e3: x + y
     

— Operator: ^ expression exponent

The infix operator ^ is used for exponentiation of scalar quantitites or for componentwise exponentiation of bunches. For example, 2^5 returns 32. Unlike the other scalar infix operators, one cannot use ^ for component-wise operations on bunches. Furthermore, one should not try to use ^ to raise a square matrix to a power. Instead, one should use ^^.

          e7 : (1+x)^4;
          
                           2      3    4
          e7: 1 + 4 x + 6 x  + 4 x  + x
     

— Operator: = expression1 expression2

In Jacal, the equals sign = is not used for conditionals and it is not used for assignments. To assign one value to another, use either : or :=. The operator = merely returns a value of the form 0 = expression. The value returned by a = b, for example is 0 = a - b.

          e6 : 1=2;
          
          e6: 0 = -1
     

— Operator: || Z1 Z2

The infix operator || is from electrical engineering and represents the effective impedance of the parallel connection of components of impedances Z1 and Z2:

          e1 : Z1 || Z2;
          
               Z1 Z2
          e1: -------
              Z1 + Z2
     

— Command: eliminate [eqn_1 eqn_2 ...] [var_1 var_2 ...]

Here eqn_i is an equation for i = 1 ... n and where var_j is a variable for j = 1 ... m. eliminate returns a list of equations obtained by eliminating the variables var_1, ..., var_m from the equations eqn_1, ..., eqn_n.

          e39 : eliminate([x^2+y=0,x^3+y=0],[x]);
          
                          2
          e39: 0 = - y - y
          
          e40 : eliminate([x+y+z=3,x^2+y^2+z^2=3,x^3+y^3+z^3=3],[x,y]);
          
          e40: 0 = 1 - z
     

— Command: suchthat var eqn

The equation eqn must contain an occurence of variable var. suchthat returns an expression for all complex values of var satisfying eqn. suchthat is useful for extracting an expression from an equation.

          e0 : a*x+b*y+c = 0;
          
          e0: 0 = c + a x + b y
          
          e1 : suchthat(x, e0);
          
              - c - b y
          e1: ---------
                  a
     

— Command: suchthat var exp

If an expression rather than an equation is given to suchthat, it is as though the equation exp=0 was given.

          e2 : suchthat(x, e0);
          
              - c - b y
          e2: ---------
                  a
     

— Operator: | var exp_or_eqn

An alternative infix notation is also available for suchthat.

When used in combination with the `{ }' notation for or, the set notation used by some textbooks results.

If var in eqn has multiple roots, a named field extension will be introduced to represent any one of those roots. When multiple values are returned, the result (in disp2d and standard grammars) is wrapped with `{ }'.

          e3 : x | a*x^2 + b*x + c;
          
                                         2
          ext3: {:@ | 0 = c + b :@ + a :@ }
          e3: ext3
          
          e4 : e3 ^ 2;
          
              - c - b ext3
          e4: ------------
                   a
     

— Command: extrule extsym

Returns the rule defining named field extension extsym.

          e5 : extrule(ext3);
          
                                     2
          e5: 0 = c + b ext3 + a ext3
     

— Command: or expr_1 ...
— Command: or eqn_1 ...

The function or takes as inputs one or more equations or values. If the inputs are equations, then or returns an equation which is equivalent to the assertion that at least one of the input equations holds. If the inputs to or are values instead of two equations, then the function or returns a multiple value. If the inputs to or consist of both equations and values, then or will return the multiple values.

          e1 : or(x=2,y=3);
          
          e1: 0 = -6 + 3 x + (2 - x) y
          
          e2 : or(2,3);
          
                                      2
          e2: {:@ | 0 = -6 + 5 :@ - :@ }
          
          e3 : e2^2;
          
                                        2
          e3: {:@ | 0 = -36 + 13 :@ - :@ }
          
          e4 : or(x=2,17);
          
          e4: 17
     

`{eqn, ... }' can be used as an alternate syntax for or:

          e5 : {+1, -1};
          
                               2
          e5: {:@ | 0 = -1 + :@ }
     

— Command: num expr

The function num takes a rational expression as input and returns a numerator of the expression.

          e25 : num((x^2+y^2)/(x^2-y^2));
          
                  2    2
          e25: - x  - y
          
          e26 : num(7/4);
          
          e26: 7
          
          e27 : num(7/(4/3));
          
          e27: 21
     

— Operator: denom rational-expression

The Jacal command denom is used to obtain the denominator of a rational expression.

          e26 : denom(4/5);
          
          e26: 5
     

— Command: listofvars expr

The command listofvars takes as input a rational expression and returns a list of the variables that occur in that expression.

          e7 : listofvars(x^2+y^3);
          
          e7: [x, y]
          
          e8 : listofvars((x^2+y^3)/(2*x^7+y*x+z));
          
          e8: [z, x, y]
     

— Command: imagpart z

Returns the coefficient of %i in expression z;

— Command: realpart z

Returns all but the coefficient of %i in expression z;

— Command: abs z
— Command: cabs z

| z |

Returns the square root of the sum of the squares of the realpart and the imagpart of z.

— Operator: degree poly var

Returns the degree of polynomial or equation poly in variable var. — Operator: degree poly

Returns the total-degree, the degree of its highest degree monomial, of polynomial or equation poly.

          e26 : degree(a*x*x + b*y*x + c*y*y + d*x + e*y + f, y);
          
          e26: 2
          
          e27 : degree(a*x*x + b*y*x + c*y*y + d*x + e*y + f);
          
          e27: 3
     

— Operator: coeff poly var
— Operator: coeff poly var deg
— Operator: coeffs poly var

The command coeff is used to determine the coefficient of a certain power of a variable in a given polynomial. Here poly is a polynomial and var is a variable. If the optional third argument is omitted, then Jacal returns the coefficient of the variable var in poly. Otherwise it returns the coefficient of var^deg in poly. The function coeffs returns a list of all of the coefficients. For example,

          e14 : coeff((x + 2)^4, x, 3);
          
          e14: 8
          
          e15 : (x + 2)^4;
          
                               2      3    4
          e15: 16 + 32 x + 24 x  + 8 x  + x
          
          e16 : coeff((x + 2)^4, x);
          
          e16: 32
          
          e18 : coeffs((x + 2)^4, x);
          
          e18: [16, 32, 24, 8, 1]
     

— Operator: poly var vect
— Operator: poly var coeff1 ...

The function poly provides an inverse to the function coeffs, allowing one to recover a polynomial from its vector or list of coefficients.

          e15 : poly(y, [16, 32, 24, 8, 1]);
          
                               2      3    4
          e15: 16 + 32 y + 24 y  + 8 y  + y
          
          e16 : poly(y, 16, 32, 24, 8, 1);
          
                               2      3    4
          e16: 16 + 32 y + 24 y  + 8 y  + y
     

— Operator: poly eqn

The function poly returns the expression equal to 0 in equation eqn. Be aware that the sign and scaling of the returned polynomial will not necessarily match those in the equation creating eqn.

          e17 : 2*a = 4*c;
          
          e17: 0 = - a + 2 c
          
          e18 : poly(e17);
          
          e18: - a + 2 c
     

— Operator: content poly var

Returns a list of content and primitive part of a polynomial with respect to the variable. The content is the GCD of the coefficients of the polynomial in the variable. The primitive part is poly divided by the content.

          e24 : content(2*x*y+4*x^2*y^2,y);
          
                               2
          e24: [2 x, y + 2 x y ]
     

— Operator: divide dividend divisor var
— Operator: divide dividend divisor

The command divide treats divident and divisor as polynomials in the variable var and returns a pair `[quotient, remainder]' such that dividend = divisor * quotient + remainder. If the third argument var is omitted Jacal will choose a variable on its own with respect to which it will do the division. In particular, of dividend and divisor are both numerical, one can safely omit the third argument.

          e5 : divide(x^2+y^2,x-7*y^2,x);
          
                      2    2       4
          e5: [x + 7 y , y  + 49 y ]
          
          e6 : divide(-7,3);
          
          e6: [-2, -1]
          
          e11 : divide(x^2+y^2+z^2,x+y+z);
          
                                 2              2
          e11: [- x - y + z, 2 x  + 2 x y + 2 y ]
          
          e14 : divide(x^2+y^2+z^2,x+y+z,y);
          
                                 2              2
          e14: [- x + y - z, 2 x  + 2 x z + 2 z ]
          
          e15 : divide(x^2+y^2+z^2,x+y+z,z);
          
                                 2              2
          e15: [- x - y + z, 2 x  + 2 x y + 2 y ]
     

— Command: mod poly1 eqn var
— Command: mod poly1 poly2 var
— Command: mod poly1 poly2

Returns poly1 reduced with respect to poly2 (or eqn) and var. If poly2 is univariate, the third argument is not needed. — Command: mod poly1 n

Returns poly1 with all the coefficients taken modulo n. — Command: mod poly1

Returns poly1 with all the coefficients taken modulo the current modulus.

If the modulus (n or the current modulus) is negative, then the results use symmetric representation.

          e19 : x^4+4 mod 3;
          
                    4
          e19: 1 + x
          
          e20 : x^4+4 mod x^2=2;
          
          e20: 8
          
          e22 : mod(x^3*a*7+x*8+34, -3);
          
                          3
          e22: 1 - x + a x
          
          e23 : mod(5,2);
          
          e23: 1
          
          e24 : mod(x^4+4,x^2=2,x);
          
          e24: 8
     

— Command: gcd poly_1 poly_2

The Jacal function gcd takes as arguments two polynomials with integer coefficients and returns a greatest common divisor of the two polynomials. This includes the case where the polynomials are integers.

          e1 : gcd(x^4-y^4,x^6+y^6);
          
               2    2
          e1: x  + y
          
          e2 : gcd(4,10);
          
          e2: 2
     

— Command: discriminant poly var

Here poly is a polynomial and var is a variable. This function returns the square of the product of the differences of the roots of the polynomial poly with respect to the variable var.

          e7 : discriminant(x^3 - 1, x);
          
          e7: -27
     

— Command: resultant poly_1 poly_2 var

The function resultant returns the resultant of the polynomials poly_1 and poly_2 with respect to the variable var.

          e2 : resultant(x^2 + a, x^3 + a, x);
          
               2    3
          e2: a  + a
     

— Command: equatecoeffs z1 z2 var

Returns the list of equations formed by equating each coefficient of variable var^n in z1 to the corresponding coefficient of var^n in z2. z1 and z2 can be polynomials or ratios of polynomials.

— Command: factor int

The Jacal command factor takes as input an integer and returns a list of the prime numbers that divide it, each occurring with the appropriate multiplicity in the list. If the number is negative, the list will begin with -1.

The results of the factor command are shown in a special factored format, which appears as the product of the factors.

          e0 : factor(120);
          
               3
          e0: 2  3 5
          
          e1 : factor(-120);
          
                  3
          e1: -1 2  3 5
     

— Command: factor polyratio

Given a univariate ratio of polynomials polyratio, returns a matrix of factors and exponents.

As above, the results are shown in factored form.

          e2 : factor((14*x^4-10/68*x^-5)/(5*x^2+1));
          
                           9
                 -5 + 476 x
          e2: ------------------
                           2   5
              2 17 (1 + 5 x ) x
          
          e3 : (14*x^4-10/68*x^-5)/(5*x^2+1);
          
                         9
               -5 + 476 x
          e3: --------------
                  5        7
              34 x  + 170 x
          
          e4 : (476*x^9-5)/(34*(5*x^2+1)*x^5);
          
                         9
               -5 + 476 x
          e4: --------------
                  5        7
              34 x  + 170 x
     

          e5 : factor(x*y);
          
          e5: y x
          
          e6 : factor((x+a)*(y^4-z));
          
                             4
          e6: -1 (a + x) (- y  + z)
          
          e7 : factor((x+u*a^3)*(y^4-z));
          
                   3            4
          e7: -1 (a  u + x) (- y  + z)
          
          e8 : factor((x+u*a^3)^2*(y^4-z)/((x+1)*(u^2-v^2)));
          
                   4        3       2
               (- y  + z) (a  u + x)
          e8: -------------------------
              (1 + x) (- u + v) (u + v)
          
          e9 : factor(200*(-1*x+1+y)*(u-r^6)*(21*x+2-t^4));
          
               3  2     6                        4
          e9: 2  5  (- r  + u) (1 - x + y) (2 - t  + 21 x)
          
          e10 : factor(2*(a+u)*(-v+b)*(a*x+y)^2);
          
                                               2
          e10: -1 2 (a + u) (- b + v) (a x + y)
          
          e11 : factor(2*(a+u)*(-v+b)*(a*x+y)^2/((u^2-v^2)*(11*x+55)));
          
                                            2
               2 (a + u) (- b + v) (a x + y)
          e11: ------------------------------
                11 (5 + x) (- u + v) (u + v)
          
          e12 : factor(2*(a+u)*(-v+b)*(a*x+y)^2/((u^2-v^2)*x^4*(11*x+55)));
          
                                            2
               2 (a + u) (- b + v) (a x + y)
          e12: -------------------------------
                                             4
               11 (5 + x) (- u + v) (u + v) x
          
          e13 : factor((c^3*u+b*a)*(b*b*a+v*p^2*q^2*c));
          
                       3        2      2  2
          e13: (a b + c  u) (a b  + c p  q  v)
          
          e14 : factor((2*z+y-x)*(y^3-a*x^2)*(b*z^2+y));
          
                                       2        2    3
          e14: (- x + y + 2 z) (y + b z ) (- a x  + y )
          
          e15 : factor((a*a*b*z+d)*(2*a*b*b*z+c));
          
                     2                2
          e15: (d + a  b z) (c + 2 a b  z)
          
          e16 : factor((a*a*b*z+d)*(2*a*b*b*z+c)*((u+a)*x+1));
          
                                     2                2
          e16: (1 + (a + u) x) (d + a  b z) (c + 2 a b  z)
          
          e17 : factor((c*z+a)*(a*z+b)*(b*z+c));
          
          e17: (b + a z) (c + b z) (a + c z)
          
          e18 : factor((a*a*b*(x+w)*z+d)*(2*a*b*b*z+c));
          
                      2        2                   2
          e18: (d + (a  b w + a  b x) z) (c + 2 a b  z)
          
          e19 : factor(((x+w)^2*z-u*d)*(-2*a*b*z+c));
          
                                             2            2
          e19: -1 (- c + 2 a b z) (- d u + (w  + 2 w x + x ) z)
          
          e20 : factor((-200*%i*x-c)*(x-d-z^5)/(a*(b^3-(a+u)*z)));
          
                                           5
               -1 (c + 200 %i x) (d - x + z )
          e20: ------------------------------
                          3
                    a (- b  + (a + u) z)
     

— Command: sff poly

Given a primitive univariate polynomial poly, calculate the square free factorisation of poly. A primitive polynomial is one with no factors (other than units) common to all its coefficients.

— Command: ffsff poly p
— Command: ffsff poly p m

Given a monic polynomial poly, a prime p, and a positive integer m, calculate the square free factorisation of poly in GF(p^m)[x]. If m is not supplied, 1 is assumed.

          e0 : ffsff(x^5+x^3+1, 53);
          
              [                2    3   ]
              [16 - 22 x + 26 x  + x   1]
          e0: [                         ]
              [       -13 + x          2]
     

— Command: berl poly n

Given a square-free univariate polynomial poly and an integer power of a prime, q, returns (as a bunch) the irreducible factors of poly.

          e2 : berl(x^5+x^3+2, 53);
          
                                  2               2
          e2: [1 + x, 5 - 26 x + x , 11 + 25 x + x ]
     

— Command: parfrac polyratio

Returns the partial fraction expansion of a rational univariate polynomial polyratio. The denominator of polyratio must be square free. This code is still being developed.

— Operator: differential expr
— Operator: ' expr

The Jacal command differential computes the derivative of the expression expr with respect to a generic derivation. It is generic in the sense that nothing is assumed about its effect on the individual variables. The derivation is denoted by a right quote.

          e6 : differential(x^2+y^3);
          
                          2
          e6: 2 x x' + 3 y  y'
          
          e7 : (x^2+y^3)';
          
                          2
          e7: 2 x x' + 3 y  y'
     

— Command: diff expr var1 ...

The Jacal command diff computes the derivative of the expression expr with respect to var1, ....

          e6 : diff(x^2+y^3,y);
          
                 2
          e6: 3 y
     

— Command: partial expr var1 ...

The Jacal command partial computes the partial derivative of the expression expr with respect to var1, ....

          e6 : partial(x^2+@1^3,1);
          
                  2
          e6: 3 @1
     

— Command: PolyDiff poly var1 ...

The Jacal command PolyDiff computes the derivative of the expression poly with respect to var1, .... It is faster than diff but poly must be a polynomial.

— Operator: bunch elt_1 elt_2 ...

[elt_1, elt_2, ...]

To collect any number of Jacal objects into a bunch, simply enclose them in square brackets. For example, to make the bunch whose elements are 1, 2, 4, type [1, 2, 4]. One can also nest bunches, for example, [1, [[1, 3], [2, 5]], [1, 4]]. Note however that the bunch whose only element is [1, 2, 3] is [1 2 3]. It is importance to notice that one has commas and the other doesn't.

          e3 : a:bunch(1, 2, 3);
          
          e3: [1, 2, 3]
          
          e4 : b:[a];
          
          e4: [1  2  3]
          
          e5 : c:[b];
          
          e5: [[1, 2, 3]]
          
          e6 : [[[1, 2, 3]]];
          
          e6: [[1, 2, 3]]
     

— Operator: flatten bnch

Removes bunch nesting from bnch, returning a single bunch of the constituent expressions and equations.

          e0 : flatten([a, [b, [c, d]], [5]]);
          
          e0: [a, b, c, d, 5]
     

— Command: ident n

The command ident takes as argument a positive integer n and returns an nxn identity matrix. This is sometimes more convenient than obtaining this same matrix using the command scalarmatrix.

          e6 : ident(4);
          
              [1  0  0  0]
              [          ]
              [0  1  0  0]
          e6: [          ]
              [0  0  1  0]
              [          ]
              [0  0  0  1]
     

— Command: scalarmatrix size entry

The command scalarmatrix takes as inputs a positive integer size and an algebraic expression entry and returns an n * n diagonal matrix whose diagonal entries are all equal to entry, where n = size.

          e1 : scalarmatrix(3, 6);
          
              [6  0  0]
              [       ]
          e1: [0  6  0]
              [       ]
              [0  0  6]
     

— Command: diagmatrix list

The Jacal command diagmatrix takes as input a list of objects and returns the diagonal matrix having those objects as diagonal entries. In case one wants all of the diagonal entries to be equal, it is more convenient to use the command scalarmatrix.

          e3 : diagmatrix(12,3,a,s^2);
          
              [12  0  0  0 ]
              [            ]
              [0   3  0  0 ]
          e3: [            ]
              [0   0  a  0 ]
              [            ]
              [0   0  0   2]
              [          s ]
          
          e4 : diagmatrix([1,2],2);
          
              [[1, 2]  0]
          e4: [         ]
              [  0     2]
     

— Command: sylvester poly_1 poly_2 var

Here, poly_1 and poly_2 are polynomials and var is a variable. The function sylvester returns the matrix introduced by Sylvester (A Method of Determining By Mere Inspection the Derivatives from Two Equations of Any Degree, Phil.Mag. 16 (1840) pp. 132-135, Mathematical Papers, vol. I, pp. 54-57) for computing the resultant of the two polynomials poly_1 and poly_2 with respect to the variable var. If one wants to compute the resultant itself, one can simply use the command resultant with the same syntax.

          e5 : sylvester(a0 + a1*x + a2*x^2 + a3*x^3, b0 + b1*x + b2*x^2, x);
          
              [a3  a2  a1  a0  0 ]
              [                  ]
              [0   a3  a2  a1  a0]
              [                  ]
          e5: [b2  b1  b0  0   0 ]
              [                  ]
              [0   b2  b1  b0  0 ]
              [                  ]
              [0   0   b2  b1  b0]
     

— Command: genmatrix function rows cols

The function genmatrix takes as arguments a function of two variables and two positive integers, rows and cols. It returns a matrix with the indicated numbers of rows and columns in which the $(i,j)$th entry is obtained by evaluating function at $(i,j)$. The function may be defined in any of the ways available in Jacal, i.e previously by an explicit algebraic definition, by an explicit lambda expression or by an implicit lambda expression.

          e4 : @1^2+@2^2;
          
                                 2      2
          e4: lambda([@1, @2], @1  + @2 )
          
          e5 : genmatrix(e4,3,5);
          
              [2   5   10  17  26]
              [                  ]
          e5: [5   8   13  20  29]
              [                  ]
              [10  13  18  25  34]
     

— Command: row matrix i

The command row returns the ith row of the matrix matrix, where i = int. If int is larger than the number of rows of matrix, then Jacal prints an error message. The corresponding command for columns of a matrix is col.

          e3 : u:[[1, 2, 3], [1, 5, 3]];
          
              [1  2  3]
          e3: [       ]
              [1  5  3]
          
          e4 : row(u, 2);
          
          e4: [1, 5, 3]
     

— Command: col matrix integer

The command col is used to extract a column of a matrix. Here, matrix is a matrix and integer is a positive integer. If that integer exceeds the number of columns, an error message such as

          ERROR: list-ref: Wrong type in arg1 ()
     

appears. Here is an example of correct use of the command col:

          e19 : a:[[1,2,4],[2,5,6]];
          
               [1  2  4]
          e19: [       ]
               [2  5  6]
          
          e20 : col(a,2);
          
               [2]
          e20: [ ]
               [5]
     

— Command: minor matrix i j

The command minor returns the submatrix of matrix obtained by deleting the ith row and the jth column.

          e21 : b:[[1,2,3],[3,1,5],[5,2,7]];
          
               [1  2  3]
               [       ]
          e21: [3  1  5]
               [       ]
               [5  2  7]
          
          e22 : minor(b,3,1);
          
               [2  3]
          e22: [    ]
               [1  5]
     

— Command: cofactor matrix i j

The command cofactor returns the determinant of the i, j minor of matrix.

— Command: rapply bunch int_1 int_2 ...

The function rapply is used to access elements of bunches. It can also access elements nested at lower levels in a bunch. In particular, it can also access matrix elements. In the above syntax, bunch is the bunch whose parts one wishes to access, and n, int_1, int_2, ..., int_n are positive integers. It returns the int_n-th element of the int_{n-1}-th element of ... of the int_2-th element of the int_1-th element of bunch. One can have n = 0. In that case, rapply simply returns the bunch.

          e2 : rapply([[1,2,3],[1,4,6],3],2,3);
          
          e2: 6
          
          e6 : rapply([a,b],2);
          
          e6: b
          
          e7 : rapply([a,b]);
          
          e7: [a, b]
     

— Command: . matrix1 matrix2

Matrix multiplication.

          e1 : a:[[1, 2, 3], [5, 2, 7]];
          
              [1  2  3]
          e1: [       ]
              [5  2  7]
          
          e2 : b:[[3, 2], [6, 4]];
          
              [3  2]
          e2: [    ]
              [6  4]
          
          e3 : b . a;
          
              [13  10  23]
          e3: [          ]
              [26  20  46]
     

— Command: ^^ matrix exponent

The infix operator ^^ is used for raising a square matrix to an integral power.

          e8 : a:[[1, 0], [-1, 1]];
          
              [1   0]
          e8: [     ]
              [-1  1]
          
          e9 : a^^3;
          
              [1   0]
          e9: [     ]
              [-3  1]
     

Negative exponents raise the inverse matrix to a power.

          e8 : [[a, b], [c, d]];
          
              [a  b]
          e8: [    ]
              [c  d]
          
          e9 : e8^^-1;
          
              [     d           - b    ]
              [-----------  -----------]
              [- b c + a d  - b c + a d]
              [                        ]
          e9: [    - c           a     ]
              [-----------  -----------]
              [- b c + a d  - b c + a d]
          
          e10 : e8^^-2;
          
               [               2                  - a b - b d       ]
               [        b c + d            -------------------------]
               [-------------------------   2  2                2  2]
               [ 2  2                2  2  b  c  - 2 a b c d + a  d ]
               [b  c  - 2 a b c d + a  d                            ]
               [                                    2               ]
          e10: [       - a c - c d                 a  + b c         ]
               [-------------------------  -------------------------]
               [ 2  2                2  2   2  2                2  2]
               [b  c  - 2 a b c d + a  d   b  c  - 2 a b c d + a  d ]
          
          e11 : e8 . e9;
          
               [1  0]
          e11: [    ]
               [0  1]
          
          e12 : e9 . e8;
          
               [1  0]
          e12: [    ]
               [0  1]
          
          e13 : e10 . e8 . e8;
          
               [1  0]
          e13: [    ]
               [0  1]
     

— Command: dotproduct vector_1 vector_2

The Jacal function dotproduct returns the dot product of two row vectors of the same length. It will also give the dot product of two matrices of the same size by computing the sum of the dot products of the corresponding rows or, what is the same, the trace of one matrix times the transpose of the other one.

          e28 : a:[1,2,3]; b:[3,1,5];
          
          e28: [1, 2, 3]
          
          e29 :
          e29: [3, 1, 5]
          
          e30 : dotproduct(a,b);
          
          e30: 20
     

— Command: crossproduct vector_1 vector_2

The Jacal command crossproduct computes the cross product of two vectors. By definition, the two vectors must each have three components.

          e24: [2 x, y + 2 x y ]
          
          e25 : crossproduct([1,2,3],[4,2,5]);
          
          e25: [4, 7, -6]
     

— Command: determinant matrix

The Jacal command determinant computes the determinant of a square matrix. Attempting to take the determinant of a non-square matrix will produce an error message.

          e1 : a:[[1,2],[6,7]];
          
              [1  2]
          e1: [    ]
              [6  7]
          
          e2 : determinant(a);
          
          e2: -5
     

— Command: transpose matrix

Computes the transpose of (matrix).

— Command: tmult matrix_1 matrix_2 index_1 index_2

tmult takes a minimum of two arguments which are the tensors on which the multiplication operation is to be performed.

With no additional arguments, tmult will produce the outer product of the two input tensors. The rank of the resulting tensor is the sum of the inputs' ranks, and the components of the result are formed from the pair-wise products of components of the inputs. For example, for the input tensors x[a,b] and y[c]

          z:tmult(x,y);=> z[a,b,c] = x[a,b]*y[c]
     

With an additional argument, tmult will produce the inner product of the two tensors on the specified index. For example, given x[i,j] and y[k,l,m]

          z:tmult(x,y,3);
          =>
                                  length
                                  -----
                                  \
                       z[a,b,c] =  >   x[a,q] * y[b,c,q]
                                  /
                                  -----
                                  q = 1
     

Note that in this case x only has 2 indices. All of JACAL's tensor operations modify index inputs to be between 1 and the rank of the tensor. Thus, in this example, the 3 is modified to 2 in the case of x. As another example, with x[i,j,k] and y[l,m,n]

          z:tmult(x,y,2);
          =>
                                    length
                                    -----
                                    \
                       z[a,b,c,d] =  >   x[a,q,b] * y[c,q,d]
                                    /
                                    -----
                                    q = 1
     

With four arguments, tmult produces an inner product of the two tensors on the specified indices. For example, for x[i,j] and y[k,l,m]

          z:tmult(x,y,1,3);
          =>
                                  length
                                  -----
                                  \
                       z[a,b,c] =  >   x[q,a] * y[b,c,q]
                                  /
                                  -----
                                  q = 1
     

Note that matrix multiplication is the special case of an inner product (of two "two dimensional matrices") on the second and first indices, respectively: tmult(x,y,2,1) == ncmult(x,y)

Finally, tmult handles the case of a scalar times a tensor, in which case each component of the tensor is multiplied by the scalar.

— Command: contract matrix index1 ...

The contraction operation produces a tensor of rank two less than a given tensor. It does this by performing a summation over two of the indices of the given tensor, as clarified in the examples below.

contract takes at least one argument which is the tensor on which the contraction operation is to be performed. One or two additional arguments may be provided to specify the indices to be used in the summation. If no additional arguments are provided, the summation is performed over the first and second indices. With one additional argument, the summation is over the specified index and the one following it (e.g., if 3 is specified, the third and fourth indices are used). With two additional arguments, the summation is performed over the indices specified. The actual indices used will be constrained to be between 1 and the rank of the tensor.

Examples:

1) For a square matrix (tensor of rank 2), contract returns a scalar that is the sum of the diagonal elements of the matrix.

2) Given x[i,j,k,l], the command

          y:contract(x,2,4);
     

produces:

                                  length
                                  -----
                                  \
                         y[a,b] =  >   x[a,q,b,q]
                                  /
                                  -----
                                  q = 1
     

Special cases: If contract is given a scalar (rank 0 tensor) as input, it just returns the scalar. For a vector (tensor of rank 1), contract returns a scalar that is the sum of the elements of the vector.

— Command: indexshift matrix index1 ...

indexshift rearranges the indices of a tensor. It is one of two generalizations of the matrix transpose operation (cf. indexswap).

indexshift takes at least one argument which is the tensor on which the index shifting is to be performed. One or two additional arguments may be provided to specify the index and the position to which it is to be shifted. If no additional arguments are provided, the first index of the tensor is shifted to the second position (equivalent the matrix transpose operation). If one additional argument is provided, it specifies the index to be shifted, and that index will be shifted "to the right" one position (e.g., if 3 is specified, the third index will be shifted to the forth position). If two additional arguments are provided, the first specifies the index and the second specifies the position to which it is to be shifted. The actual index shifted and its shifted position will be constrained to be between 1 and the rank of the tensor.

For example, given x[a,b,c,d], the command y:indexshift(x,1,3); produces a tensor y such that y[a,b,c,d] == x[b,c,a,d]. In this example, the element that was in position [a,b,c,d] in x will be in position [b,c,a,d] in y.

Special cases: If indexshift is given a scalar (rank 0 tensor) as input, it just returns the scalar. For a vector (tensor of rank 1), indexshift transposes the 1-by-n matrix (row vector) to an n-by-1 matrix (column vector).

— Command: indexswap tensor ...

indexswap rearranges the indices of a tensor. It is one of two generalizations of the matrix transpose operation (cf. indexshift).

indexswap takes at least one argument which is the tensor on which index swapping is to be performed. One or two additional arguments may be provided to specify the indices to be swapped. If no additional arguments are provided, the first and second indices of the tensor are swapped (equivalent the matrix transpose operation). With one additional argument, the specified index is swapped with the one following it (e.g., if 2 is specified, the second and third indices will be swapped). If two additional arguments are provided, they specify the indices to be swapped. The actual indices used will be constrained to be between 1 and the rank of the tensor.

For example, given x[a,b,c,d], the command y:indexswap(x,2,4); produces a tensor y such that y[a,b,c,d] = x[a,d,c,b]. In this example, the element that was in position [a,b,c,d] in x will be in position [a,d,c,b] in y.

Special cases: If indexswap is given a scalar (rank 0 tensor) as input, it just returns the scalar. For a vector (tensor of rank 1), indexswap transposes the 1-by-n matrix (row vector) to an n-by-1 matrix (column vector).

— Operator: lambda varlist expression

Jacal has the ability to work with lambda expressions, via the command lambda. Furthermore, Jacal always converts user definitions of functions by any method into lambda expressions and converts the dummy variables of the function definition into symbols such as 1, 2, .... Jacal can manipulate lambda expressions by manipulating their function parts, as in `e14' below. Jacal can also invert a function using the command finv.

          e12 : lambda([x],x^2);
          
                              2
          e12: lambda([@1], @1 )
          
          e13 : lambda([x,y,z],x*y*z);
          
          e13: lambda([@1, @2, @3], @1 @2 @3)
          
          e14 : e12+e13;
          
                                      2
          e14: lambda([@1, @2, @3], @1  + @1 @2 @3)
     

— Command: finv function

function^^-1

The command finv takes as input a function of one variable and returns the inverse of that function. The function may be defined in any of the ways permitted in Jacal, i.e. by an explicit algebraic definition, by an explicit lambda expression or by an implicit lamba expression. If f is the function, then typing f^^-1 has the same effect as typing finv(f).

          e0 : w(t):=t+1;
          
          w(t): lambda([@1], 1 + @1)
          
          e0 : finv(w);
          
          e0: lambda([@1], -1 + @1)
     

— Command: %

The symbol % represents the last expression obtained by Jacal. It can be used in formulas like any other constant or variable or expression.

          e21: 5
          
          e22 : %;
          
          e22: 5
          
          e23 : %^2;
          
          e23: 25
     

— Command: batch filename

The command batch is used to read in a file containing programs written in Jacal. Here, filename is a string in double quotes. The precise way in which one refers to a file is, of course, system dependent.

          batch("demo");
     

of the file demo in the JACAL directory will give a demonstration of JACAL's capabilities.

— Command: commands

The command commands produces a list of all of the command available in Jacal. It is called as s function of no arguments. Explicitly:

          e21 : commands();
          
               u-/+ u+/- transpose transcript differential terms system
               sylvester show set scalarmatrix row resultant rapply quit
               qed discriminant poly or num negate mod minor matrix
               load listofvars ident genmatrix gcd finv factor example
               eliminate dotproduct divide diff diagmatrix determinant
               describe denom crossproduct content commands col coeffs
               coeff bunch batch b+/- ^^ ^ = / - + * %
     

— Command: describe command

The command describe is the heart of the online help facility of Jacal. Here, command is a string which is the name of a command and describe produces a brief description of the command and in many cases includes an example of its use. Together with the command commands(), which prints a list of all available Jacal commands, and the command example, which gives an example of the use of the command, one can in principle use Jacal without a manual after one has learned how to get started.

          e27 : describe(col);
          column.  column of a matrix
          e27 : describe(resultant);
          resultant.  The result of eliminating a variable between 2
          equations (or polynomials).
          27 : describe(+);
          Addition, plus.
          a + b
     

— Command: example command

Here, command is a string which is the name of a Jacal command. example gives an example of the use of the command. See also describe.

          e43 : example(+);
          a + b
          
          e43: a + b
     

— Command: load string

The Jacal command load takes as input a string and reads in a `Scheme' file whose name is obtained by appending the extension .scm to the string. If you want to read in a file of Jacal commands, do not use load. Instead use the command batch. To load in the file foo.scm,

          e9 : load("foo");
          
          e9: foo
     

— Command: qed

Exit from Jacal to Scheme. With interactive Scheme systems (such as SCM), It does not return you to the operating system. Instead it suspends Jacal and returns you to the underlying scheme. You can return to the Jacal session where you left off by simply typing (math). If you do not wish to return to Jacal but really want to terminate the session and return to the operating system, then after typing qed();, type (slib:exit) or use quit. — Command: quit

Exit directly from Jacal to the operating system. You will not be able to continue your Jacal session.

          type qed(); to return to scheme
          e1 : qed();
          scheme
          > (math)
          type qed(); to return to scheme
          e2 : quit();
          unix>
     

— Command: system command

One can issue commands to the operating system without leaving Jacal. To do this, one uses the command system. For example, in a UNIX operating system, the command system("ls"); will print the directory. One way in which the command system might be especially useful is to edit files containing Jacal scripts without leaving Jacal, particularly in non-UNIX machines or on machines without GNU emacs.

          e0 : system("echo hi there");
          hi there
          
          e0: 0
     

— Command: terms

Prints a copy of the GNU General Public License

          e1 : terms();
                              GNU GENERAL PUBLIC LICENSE
                                 Version 3, 29 June 2007
          
           Copyright (C) 2007 Free Software Foundation, Inc. <http://fsf.org/>
           Everyone is permitted to copy and distribute verbatim copies
           of this license document, but changing it is not allowed.
     

[ rest deleted for brevity]

— Command: transcript string

The command transcript allows one to record a Jacal session. It is called with the syntax transcript(string);, where string is the name of the file in which one wants to keep the transcript of the session. When one wishes to stop recording, one types transcript();. One is then free to use transcript again later in the session on another file. One can use it on the same file, but the file is overwritten. Presently, the command transcript does not echo commands to a file.

          e9 : a:[1,2,3];
          
          e9: [1, 2, 3]
          
          e10 : transcript("foo");
          
          e10: foo
          
          e11 : a;
          
          e11: [1, 2, 3]
          
          e12 : transcript();
          e12 : system("cat foo");
          
          e10: foo
          
          e11 : a;
          
          e11: [1, 2, 3]
          
          e12 : transcript();
          
          e12: 0
     

— Command: set flag value

There are various flags that the Jacal user can control, namely the Jacal command line prompt, the priority for printing terms in Jacal output, the input grammar and the output grammar. For a discussion of the various grammars please See Flags. The command show is closely related, allowing one to see what the current settings are.

— Command: show flag

The command show enables the Jacal user to examine the current setting of various flags as well as to list the flags that can be set by the user and to display other information. To change the settings of the flags, use the command set. To see all the information accessible through the show command, type show all. To see the available grammars, type show grammars. To see the current input grammar type show ingrammar. To see the current output grammar, type show outgrammar. To see the current priority for printing expressions, type show priority.

          e1 : show all;
          
               prompt priority outgrammar ingrammar grammars all
          e1 : show prompt;
          
          e1: e1
          
          e3 : show priority;
          
              :@ (differential :@) @3 @2 @1 %inftsl y x u-/+ u+/-
              transpose transcript differential terms t system sylvester
              showpriority show set scalarmatrix row resultant rapply quit
              qed prompt priority discriminant poly or num negate mod
              minor matrix load listofvars ident genmatrix gcd finv factor
              example eliminate e1 dotproduct divide diff diagmatrix
              determinant describe denom crossproduct content commands col
              coeffs coeff c bunch batch b-/+ b+/- b all a ^^ ^ = / - + *
              % %sqrt1 %i
          
          e3 : show outgrammar;
          
          e3: disp2d
          
          e4 : show ingrammar;
          
          e4: standard
          
          e5 : show grammars;
          
          e5: [disp2d, standard, schemepretty, scheme]
     

— Flag: prompt string

If one changes the prompt, string is a string of alphanumeric characters without quotes. After this command is executed, subsequent commands will cause new prompts to be obtained from string by incrementing it. If the prompt ends in a letter, it will be treated as a digit in base 26 and incremented. If it ends in a string of digits, that string will be treated as a number in base 10 and incremented. The remaining characters in the string will play no role in this incrementation.

          e1 : set prompt az9Z;
          e1 : a+b;
          
          az9Z: a + b
          
          az9AA : a+b;
          
          az9AA: a + b
          
          az9AB : set prompt ok99;
          az9AB : a+b;
          
          ok99: a + b
          
          ok100 : a+b;
          
          ok100: a + b
          
          ok101 :
     

— Flag: ingrammar grammar
— Flag: outgrammar grammar

The following examples show how one changes the input grammar or the output grammar.

          e1 : a:[[[1,2,3]]];
          
          e1: [[1, 2, 3]]
          
          e2 : set outgrammar standard;
          e2 : a;
          
          e2: [[[1, 2, 3]]]
          
          e3 : set outgrammar scheme;
          e3 : a;
          
          (define e3 #(#(#(1 2 3))))
          e4 : (1+x)^5;
          
          (define e4 (+ 1 (* 5 x) (* 10 (^ x 2)) (* 10 (^ x 3)) (* 5 (^ x 4)) (^ x 5)))
          e6 : set ingrammar scheme;
          e6 : (+ e4 1);
          
          (define e6 (+ 2 (* 5 x) (* 10 (^ x 2)) (* 10 (^ x 3)) (* 5 (^ x 4)) (^ x 5)))
          e7 : (set ingrammar disp2d)
          e7 : diagmatrix(3,6);
          
          (define e7 #(#(3 0) #(0 6)))
          e8 : set outgrammar disp2d;
          e8 : e7;
          
              [3  0]
          e8: [    ]
              [0  6]
          
          e9 : set outgrammar standard;
          e9 : e7;
          
          e9: [[3, 0], [0, 6]]
     

Note that in the above examples, it is possible to input and output expressions in scheme by setting the ingrammar and/or outgrammar to scheme. Doing so result in linear output (as with standard grammar) as opposed to a two dimensional display (as with disp2d). The analogue of disp2d for scheme output is scheme pretty-printing. To have such output, set the output grammar to schemepretty.

          
          e4 : set outgrammar schemepretty;
          e4 : (1+x)^5;
          
          (define e4
            (+ 1
               (* 5 x)
               (* 10 (^ x 2))
               (* 10 (^ x 3))
               (* 5 (^ x 4))
               (^ x 5)))
     

Jacal also allows for output to be automatically typeset in TeX. This can be quite useful if one wants to use the results of one's computations in published articles. Continuing with the example of (1+x)^5 above, we have:

          e5 : set outgrammar tex;
          e5 : e4;
          
          e5: 1 + 5 x + 10 x^{2} + 10 x^{3} + 5 x^{4} + x^{5}
          
          e6 : (1+1/x)^3/(1-1/y)^4;
          
          e6: {\left(1 + 3 x + 3 x^{2} + x^{3}\right) y^{4}}\over{x^{3} - 4 x^{3} y +
          
                          6 x^{3} y^{2} - 4 x^{3} y^{3} + x^{3} y^{4}}
     

— Flag: priority int

The following examples show how to set the priority of printing terms.

          e10 : a;
          
          e10: [[[1, 2, 3]]]
          
          e11 : show priority a;
          
          ;;; not a simple variable:  (((1 2 3) . ()) . ())
          
          e12 : show priority b;
          
          e12: 128
          
          e13 : show priority c;
          
          e13: 128
          
          e14 : b+c;
          
          e14: b + c
          
          e15 : c+b;
          
          e15: b + c
          
          e16 : set priority b 200;
          e16 : b+c;