JACAL MANUAL December 13, 1989 Aubrey Jaffer Bertronics 84 Pleasant St. Wakefield MA, 01880 617-245-0545 INTRODUCTION JACAL System is a computer program for the manipulation and simplification of algebraic and complex analytic equations, expressions, functions, and constants. OVERVIEW OF THE JACAL SYSTEM JACAL differs from many other symbolic mathematics systems in that the input and outupt formats are the same. Any expression typed out by JACAL can be used as input. If the user has an input editor, she can extract parts of expressions easily with it. The syntax used by JACAL is a slightly extended version of MACSYMA syntax. The extensions are the use of x' to represent differentials and the construct {x|equation}. This represents a (possibly multiple) value of the roots of equation in x. | is read as "such that". STARTING JACAL To load JACAL from unix, % cd jacal % lisp > (load "math") system dependent messages ... type (math) to begin > (math) type QED to return to lisp E1 : ENTERING EXPRESSIONS An expression now typed (for instance a-b;) will be assigned to E1. An expression can be terminated with ";" or "\$". If the end is: ";" print normalized expression "\$" print nothing As soon as the expression is parsed the label is printed again E1 As soon as the expression is succesfully evaluated and normalized a colon is printed E1 : Then the expression is factored (if desired) and printed. E1 : A - B Whenever E1 is appears in an expression from now on it is immediately replaced by its value. E2 : 4-e1 /*comments can appear anywhere a space can appear*/; E2 : 4 - A + B DEFINING SYMBOLS Other symbols can be defined by ":" or ":=" E3 : disc: b^2-4*a*c; E3 : B^2 - 4*A*C E4 : x := (-b +/-disc^(1/2))/(2*a); E4 : (- B +/- (B^2 - 4*A*C)^(1/2)) / (2*A) We could also have defined both at once: E5 : x := (-b +/-(disc: b^2-4*a*c)^(1/2))/(2*a); E5 : (- B +/- (B^2 - 4*A*C)^(1/2)) / (2*A) X is now defined the same as E4 (and DISC as E3). A symbol can be redefined, but not using the previous value of that symbol. Therefore, to reset a symbol simply define it to itself. E6 : x := x; E6 : X DEFINING FUNCTIONS E7 : f(x,y) := x^2+y^2; E7 : \@1,@2; @1^2 + @2^2 E8 : f(8,x); E8 : 64 + x^2; We could have equivalently typed E9 : f : \x,y;x^2+y^2; E9 : \@1,@2; @1^2 + @2^2 with the same effect. Functions can be added, subtracted, ... like other expressions. E10 : f*f+b+3; E10 : \@1,@2; 3 + B + @1^4 + 2*@1^2*@2^2 + @2^4 @1 and @2 are arguments while B and 3 are not. OPERATORS Consider the differential operator (a flag can be set to have output use %d instead of ') E11 : y*%d/%d(x)-x*%d/%d(y); E11 : \@1; Y*@1'/X' - X*@1'/Y' E12 : e11(x^3+y^2); E12 : (3*X^2 - 2*X)*Y EQUATIONS E7 : c=-a*x^2-b*x; E7 : C + B*X + A*X^2=0 E7 is the only symbol defined here. It's value is the entire equation. An equation can be viewed as a constraint on variables or as describing a surface in the space of the variables. An equation has no effect unless it is explicitly an argument. E8 : eliminate([e7, b^2=2*(c^2+a^2)],b); E8 : Vectors and Matrices can be represented using a BUNCHes: E11 : [[a, b, c], [d, e, f], [g, h, i]]; E11 : [[A, B, C] ,[D, E, F] ,[G, H, I]] E12 : transpose(e11); E12 : [[A, D, G] ,[B, E, H] ,[C, F, I]] E13 : determinant(%); E13 : (- C*E + B*F)*G + (C*D - A*F)*H + (- B*D + A*E)*I E49 : e11[1,2]; E49 : B E50 : e11[1]; E50 : [A, B, C] Sytems of equations can be represented in the same form. E63 : sqrtx: {y|y^2=x}; /*this defines the symbol sqrtx to be a square root of x*/ E63 : {@ | @^2 = X} E67 : sqrtx-sqrtx; E67 : 0 E68 : {y|y^2=x}-{y|y^2=x}; /* +/-sqrtx - +/-sqrtx */ E68 : {@ | @^2 = 4*X} OR 0 /*there are 3 possible answers*/ E64 : diff(sqrtx,x); /*this takes derivative of a square root of x */ E64 : X' / (2*SQRTX) /*result must be related to SQRTX */ E65 : diff({y|y^2=x},x); /*derivative of +/-sqrtx */ E65 : {@ | @^2 = X'^2 / (4*X)} E3 : sqrt(x) := x^(1/2); E3 : \@1; {@ | @^2 = @1} E4 : sqrt(y); E4 : {@ | @^2 = Y} E1 : plus + times; E1 : \@1,@2; @1 + (1 + @1)*@2 E7 : E1(4,t); E7 : 4 + 5*T Even Currying of arguments is supported: E8 : E1(q); E8 : \@2; Q + (1 + Q)*@2 E9 : E8(u); E9 : Q + (1 + Q)*U E22 : {Y | Y^2 = X^2} should reduce to: E22 : {@ | @^2 = 1}*X; Scheme translation a+b*c; (+ a (* b c)) \a,b;a+b; (lambda (a b) (+ a b)) a : b=c; (set! a (= b c)) a := g-i; (set! a (- g i)) if a=?b then c=a else c=b; the else c=b is optional (if (=? a b) (= c a) (= c b))