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SymbMath 2.2: A Symbolic Calculator with Learning (Version 2.2) by Dr. Weiguang HUANG Dept. Analytical Chemsitry, University of New South Wales, Kensington, Sydney, NSW 2033, Australia Phone: 61-2-697-4643 Fax: 61-2-662-2835 E-mail: w.huang@unsw.edu.au Copyright (C) 1990-1993 April 15, 1993 This is Part Three of SymbMath manual. The SymbMath manual has three parts in the files SymbMath.DOC, SymbMath.DO2 and SymbMath.DO3. 11. Inside SymbMath As an expert system, SymbMath consists of three major components: a knowledge base, an inference engine, and a global data base. The knowledge base is a set of rules, the inference engine is a rule interpreter for utilizing the knowledge base in the solution of the problem, and the global data base is a working memory for keeping track of the problem status, the data from the data file for the particular problem, and the solution of sub-problems. In addition, it contains a natural language interface for input and output natural languages (e.g. mathematical formulas, chemical reactions). User Library disk /|\ | /|\ | | | \|/ \|/ \|/ ------------------------------ | Natural Language Interface | ------------------------------ /|\ | \|/ ------------------------------ ------->| Inference Engine |<---------- | ------------------------------ | \|/ \|/ ------------------ -------------------- | Knowledge Base | | Global Data Base | ------------------ -------------------- /|\ | -------------------- | Data File | -------------------- Figure 11.1 Base structure of SymbMath Table 11.1 Characteristics of SymbMath -------------------------------------------------------------------- Function: Symbolic computation. Domain: Mathematics, chemistry. Search direction: Forward chaining. Control mechanism: Guessing and test, pattern match. Search space transformations: Break into sub-problems. Knowledge base representation: Rules. Developer interface: Learning, programming, library. User interface: Pull-down menu, pop-up menu, multi-windowed text editor, help, windows. System interface: numeric computation software, graphic software (e.g. PlotData), etc. Input format: Math formulas, numbers, BASIC or FORTRAN codes, chemical symbols and reactions. Output format: Math notation, BASIC or FORTRAN codes, chemical reaction equations. Input from: Keyboard, disk. Output to: Screen, disk, printer. Tool language: Prolog. Computer: IBM PC. Memory: 420 KBytes. Operating system: MS-DOS. --------------------------------------------------------------------- 12. System limits 1. The maximum character of a symbol is 64000. 2. The maximum character of an expression is 64000. 3. The range of the input real numbers is -inf, -(10^307)^(10^307) to -(10^-307)^(10^-307), 0, (10^-307)^(10^-307) to (10^307)^(10^307), inf. 4. The range of the output real numbers is the same as input when the switch Numerical=Off, but when the switch Numerical=On, it is -inf, -1.E307 to -1.E-307, 0, 1.E-307 to 1.E307, inf. 5. The maximum digit of the input numbers is 64000. 6. The maximum digit of the stored numbers is 16. 7. The maximum digit of the output numbers is 11. 8. The maximum digit of the rational numbers is 16. 9. The maximum arguments of the user-defined function with pattern is 5. 10. The maximum arguments of the user-defined rule is 2 and the maximum number of the pattern is 1. 13. Keywords SymbMath has three versions: Shareware Version A, Student Version B and Advanced Version C. The shareware version lacks the solve(), dsolve(), trig (except sin(x)), and hyerbolic functions on input, and the student version lacks the hyerbolic and dsolve() functions on input. You cannot input these lacked functions, but all versions can output all functions. Upper- and lower-case letters are different until the switch lowercase is set to on (i.e. lowercase=on). All keywords are lower-case letters until the switch lowercase is set to on. The following special symbols (about 130 words) only include the keywords in the SymbMath system and the initialization package init.sm. 13.1 Keywords in Functional Order --------------------------------------------------------------------- 1. Built-in constants: i, e, pi, inf, zero, constant, discont, undefined, complex_inf. 2. Built-in variables: last. 3. Negative and positive: -, +. 4. Algebraic operators: +, -, *, /, ^, **, (). 5. Relational operators: ==, >, >=, <, <=, <>. 6. Logic operators: and, or. 6. Algebraic functions: -x, sqrt(x), n!, fac(n), mod(x,y), div(x,y). 7. Complex functions: re(x), im(x), abs(x), arg(x), sgn(x). 8. Exponential functions: exp(x), ln(x), ei(n,x), gamma(n,x). 9. Trig functions: sin(x), cos(x), tan(x), csc(x), sec(x), cot(x), asin(x), acos(x), atan(x), acot(x), asec(x), acsc(x), atan2(x,y). 10. Hyperbolic functions: sinh(x), cosh(x), tanh(x), csch(x), sech(x), coth(x), asinh(x), acosh(x), atanh(x), acoth(x), acsch(x), asech(x). 11. Assignments: =, :=. 12. Equation: ===. 13. User-defined functions: f(x_):=x^2, f(x_):=if(isnumber(x),x^2). 14. User-defined rules: d(si(x_),x_):=sin(x)/x. 15. Calculus functions: d(y,x), d(y,x,order), d(y, x=x0), d(y, x=x0, order), inte(f,x), inte(f,x,xmin,xmax), dsolve(d(y)/d(x)===f(x,y), y), subs(y, x=x0), lim(y, x=x0), sum(f, x,xmin,xmax,dx), prod(f, x,xmin,xmax,dx), 16. Transformation functions: expand(expr), factor(expr), solve(x^2===a,x), coef(expr,x), left(x===a), right(x===a), float(x), ratio(x), round(x), trunc(x), table(p[x], x,xmin,xmax,dx). 17. Numeric computation: num(expr). 18. Test function: isodd(x), iseven(x), isinteger(x), isreal(x), isnumber(x), islist(x), isfree(y,x). 19. Table: table(f, x,xmin,xmax,dx). 20. List: [a,b], list(f, x,xmin,xmax,dx), length([a]). 21. Element of list: f[1], last[1]. 22. Switches: numerical, output, expand, expexpand, lnexpand, lowercase, =, on, off, basic, fortran, prolog, twodim. 23. Commands: system(dos), clear(x), block(), null. 24. Separators: "," , from, to, step. 25. Assume: assume(a>0), isreal(x)=1. 26. Rule: if(test, then), if(test, then, else). 27. Loop: do(expr, x,xmin,xmax,dx), repeat(expr, test). 28. Input: include('filename'). 29. Output: openfile('filename'), closefile('filename'). 30. Plot: plot(x^2,x). 31. Comment: #. 32. Action: done, cleared, assumed. -------------------------------------------------------------------- 13.2 Keywords in Alphabetical Order ---------------------------------------------------------------------- abs(x) acos(x) acosh(x) acot(x) acoth(x) acsc(x) acsch(x) arg(x) asec(x) asech(x) asin(x) asinh(x) assume(x>0) assumed atan(x) atan2(x,y) atanh(x) basic block() clear(x) cleared closefile() coef(f,x) complex_inf constant cos(x) cosh(x) cot(x) coth(x) csc(x) csch(x) d() div(x,y) discont do() done dsolve(eq,y) E e ei(n,x) end exp(x) expand(f) expand expexpand fac(x) factor(expr) float(x) fortran from gamma(n,x) i if() im(x) include('file') inf inte() iseven(x) isfree(y,x) isinteger(x) islist(x) isodd(x) isreal(x) isnumber(x) last left(x===a) length([a]) lim() list() ln(x) lnexpand lowercase mod(x,y) null num(expr) numerical off on openfile() output pi plot() prod() prolog ratio(x) re(x) repeat() right(x===a) round(x) sec(x) sech(x) sgn(x) sin(x) sinh(x) solve(eq,x) sqrt(x) subs(f,x=x0) sum() step system(dos) table() tan(x) tanh(x) to trunc(x) twodim undefined zero + - * / ^ ** () [] = := == === > >= < <= <> ! # , ----------------------------------------------------------------------- 13.3 Glossary -------------------------------------------------------------------- * abs abs(x) is the absolute value function of x. x can be complex numbers. Absolute value of the complex number x is defined by sqrt(re(x)^2+im(x)^2). The results are in the range 0 to inf. e.g. abs(-1) gives 1, abs(-i) gives -1. See also: sgn, re, im. * acos acos(x) is the arc cosine function of x. The inverse function of cos(x). The result is given in radians. See also: cos. * acosh acosh(x) is the inverse hyerbolic cosine function of x. The inverse function of cosh(x). See also: cosh. * acot acot(x) is the arc cotangent function of x. The inverse function of cot(x). The result is given in radians. acot(x)=atan(1/x). See also: cot, atan. * acoth acoth(x) is the inverse hyperbolic cotangent function of x. The inverse function of coth(x). acoth(x)=atanh(1/x). See also: coth, atanh. * acsc acsc(x) is the arc cosecant function of x. The inverse function of csc(x). The result is in radians. acsc(x)=asin(1/x). See also: csc, asin. * acsch acsch(x) is the inveres hyperbolic cosecant function of x. The inverse function of csch(x). acsch(x)=asinh(1/x). See also: csch, asinh. * arg arg(x) gives the argument of x. It gives the phase angle of x in radians. Its result is in the range > -pi to <= pi. arg(x)= atan2(re(x),im(x)). x can be complex number. For real x, arg(x)=0. See also: abs, sgn, atan2. * asec asec(x) is the arc secant function of x. The inverse function of sec(x). The result is given in radians. asec(x)=acos(1/x). See also: sec, acos. * asech asech(x) is the inverse hyperbolic secant function of x. The inverse function of sech(x). asech(x)=acosh(1/x). See also: sech, acosh. * asin asin(x) is the arc sine function of x. The inverse function of sin(x). The result is given in radians. See also: sin. * asinh asinh(x) is the inverse hyperbolic sine function of x. The inverse function of sinh(x). See also: sinh. * assume assume(x > 1), or assume(x < 1), assumes a variable x > 1, or x < 1. All variables are assumed as complex by default. e.g. assume(x>0), assume(x<0), or isreal(x) := 1. See aslo: sgn, isodd, iseven, isinteger, isreal, isnumber. * assumed assumed points out that the variable has been assumed. See also: assume. * atan atan(x) is the arc tangent function of x. The inverse function of tan(x). The result is given in radians. See also: tan. * atan2 atan2(x,y) returns the radian angle of (x,y). atan2(x,y) = sgn(y)*pi/2 if x=0 = atan(y/x) if x>0 = atan(y/x)+pi if x<0, y>=0 = atan(y/x)-pi if x<0, y<0 . Thus atan2(x,y) takes its value larger than -pi less or equal to pi. * atanh atanh(x) is the inverse hyperbolic tangent function of x. The inverse function of tanh(x). See also: tanh. * basic basic is a value of the switch output. It sets output in BASIC format. e.g. output=basic. See also: output, on, off, fortran, prolog. * block block(a,b,c) return only the last argument. e.g. block(a,b,c) gives c. * clear clear(expr) clears values and definitions for the variable, function or expression expr from memory. e.g. clear(p), clear(f(x)). * cleared It says that the variable, function or expression has been cleared from memory. See also: clear. * closefile closefile('filename') closes the file 'filename' and return the output to screen. The filename is any MS-DOS file name. See also: openfile, include. * coef coef(expr,form) gives the coefficient of form in the polynomial expr. It picks only terms that contain the particular form specified. x is not considered part of x^6. e.g. coef(2*x^6+x+4, x^6) gives 2, coef(2*x^6+x+4, x) gives 1. * complex_inf The complex infinite, both real and imaginary parts of complex numbers are infinity, as the built-in constant. complex_inf=inf+inf*i, inf-inf*i, -inf+inf*i, or -inf-inf*i. See also: inf. * constant The indefinite integral constant. * cos cos(x) is the cosine function of x. The angle x is measured in radians (multiply by degree to convert from degrees). x can be complex numbers. See also: acos, sec. * cosh cosh(x) is the hyerbolic cosine function of x. cosh(x)=(exp(x)+exp(-x))/2. x can be complex numbers. See also: acosh. * cot cot(x) is the cotangent function of x. The angle x is measured in radians. (multiply by degree to convert from degrees). cot(x)=1/tan(x). See also: acot, tan. * coth coth(x) is the hyerbolic cotangent function of x. coth(x)=1/tanh(x). See also: acoth, tanh. * csc csc (x) is the cosecant function of x. The angle x is measured in radians (multiply by degree to convert from degrees). csc(x)=1/sin(x) x can be complex numbers. See also: acsc, sin. * csch csch(x) is the hyperbolic cosecant function of x. csch(x)=1/sinh(x). x can be complex numbers. See also: acsch, sinh. * d d() gives the partial derivative. d(f, x) differentiate y with respect to x. e.g. d(x^2/d(x)) gives 2*x. d(f, x, n) gives the n-th order derivative of f with respect to an undefined variable x. d(f, x=c) gives the derivative of f with respect to an undefined variable x at x=c. d(f, x=c, n) gives the n-th order derivative of f with respect to an undefined variable x at x=c. d(y) implicit differentiation, used in differential equations, e.g. x*d(x)+y*d(y) === 0. * degree degree gives the number of radians in one degree. degree=pi/180. You can multipy by degree to convert from degree to radians. e.g. 45*degree, sin(45*degree). See also: pi. * discont The discontinuity. If the function value is discont, the function has a discontinuity and only has the one-sided limit at x=c. Users should evaluate its left-sided limit or right-sided limit by x=c-zero or x=c+zero. * do do(expr, x,xmin,xmax,dx) evaluates expr with the x looping from xmin to xmax on step dx. e.g. x=1, do(x=x+1, j,1,5,1) gives x=5. * done It indicates that the action has been done. * dsolve dsolve(d(y)/d(x)===f(x,y), y) solves the first order variables separable and linear differential equations. The d(y)/d(x) must be alone on the left hand side of the equations. e.g. dsolve(d(y)/d(x) === x*y + 1, y). See also: solve, nsolve. * E E is the exponential part of a floating point number. e.g. 1.1E2 is the same as 1.1*10^2. See also: e, exp. * e (1) e is the exponential constant (baes of natural logarithms), e=2.718..., the built-in constant, e is converted to 2.718... when the switch Numerical=On. e^x is the same as exp(x). e.g. e^2, e^x. (2) e is the exponential part of a floating point number, the same as E. e.g. 1.1e2 is the same as 1.1E2. See also: E, exp. * ei ei(n,x) is the exponential integral function En(x), ei(n,x)=inte(t^n*e^t, t,-inf,x), d(ei(n,x),x)=x^n*e^x, ei(-1, x)=ei(x), ei(0,x)=e^x. ei(x) is the exponential integral function Ei(x), ei(x) = inte(e^t/t, t,-inf,x), d(ei(x),x)=e^x/x, See also: gamma. * erf erf(x) is the error function of x. It is the probability integral function or the integral of the Gaussian distribution. erf(x)= 2/sqrt(pi)*inte(exp(-t^2),t,0,z). * exp exp(x) is the exponential function of x (base of e). The same as e^z, e=2.718... It is the inverse to ln(x). x can be complex numbers. See also: ^. * expand expand(expr) expands out products and positive powers in expr. expand works only on positive integer powers. e.g. expand((a+b)^2) gives a^2 + 2*a*b + b^2. See also: factor. the switch of expansion. expand=on e.g. c*(a+b) to c*a+c*b. expand=off disable expansion, this is default. * expexpand The switch of exponential expansion. expexpand=on e.g. c^(a+b) to c^a*c^b. expexpand=off disable exponential expansion, this is default. * fac fac(n) is the factorial function of n. The same as n!. e.g. fac(3) gives 6. See also: n!. * factor factor(expr) factorises from expr. e.g. factor(a^2 + 2*a*b + b^2) gives (a+b)^2. See also: expand, Expand, ExpExpand, LnExpand. * float float(x) converts x to the floating-point number. e.g. float(1/2) gives 0.5. See also: ratio. * fortran fortran is the value of the switch Output. It forces the output in Fortran format. e.g. output=fortran. See also: output, basic, twodim, prolog, on, off. * from The separator, the same as the comma (,). * gamma gamma(n,x) is the incomplete gamma function, gamma(n,x)= inte(t^n*e^(-t), t,0,x), d(gamma(n,x),x)=x^n*e^(-x). gamma(n,0)=0, gamma(n,inf)=gamma(n+1)=n!. gamma(n) is the gamma function Γ(n), gamma(n)=inte(t^(n-1)*e^(-t), t,0,inf). gamma(n,x) is similar to gamma(n), but its power term is t^n, instead of t^(n-1). gamma(n)=(n-1)!. See also: ei. * i i represents the imaginative unit of the complex numbers, i=sqrt(-1), as the built-in constant. e.g. 1+2*i. See also: re, im. * if if(condition, x, y) gives x if condition evaluates to 1, y if it evaluates to 0, or no output if it evaluates to neither 1 or 0. if(condition, x) gives x if condition evaluates to 1, or no output otherwise. It is useful in definition of the use-defined function to left the function unevaluted if the argument of the function is not number. e.g. define f(x_):=if(isnumber(x), 1), then call f(x), f(10) gives 1, and f(a) gives f(a). See also: :=, =. * im im(x) gives the imaginative part of the complex number x. e.g. im(1+2*i) gives 2. See also: re, abs, sgn, arg. *include include('filename') includes (or runs) the file 'filename'. The filename is any MS-DOS file name. See also: openfile, closefile. * inf inf is a positive infinity, as the built-in constant. e.g. inf+2*inf gives inf, 1/inf gives 0. See also: complex_inf. * inte The integral function. inte(f,x) find the indefinite integral of f with respect to an undefined variable x. inte(f,x,xmin,xmax) find the definite integral of f with respect to an undefined variable x taken from x=a to x=b. inte(y) implicit integration, used to integrate the differential equations. See also: ninte. * iseven iseven(x) gives 1 if x is an even integer, or 0 otherwise. You can assume x is even by iseven(x) := 1. e.g. iseven(2) gives 1, iseven(3) gives 0. See also: isodd, isinteger, islist, isreal, isnumber, isfree, sgn. * isfree isfree(y,x) gives 1 if y is free of x, or 0 otherwise. You can assume y is free of x by iseven(y,x) := 1. e.g. isfree(a*b,x) gives 1, isfree(x*y,x) gives 0. See also: isodd, isinteger, islist, isreal, isnumber, isfree, sgn. * isinteger isinteger(x) gives 1 if x is an integer, or 0 otherwise. You can assume x is integer by isinteger(x) := 1. e.g. isinteger(2) gives 1, isinteger(3.2) gives 0. See also: iseven, islist, isodd, isreal, isnumber, isfree, sgn. * islist islist(x) gives 1 if x is a list, or 0 otherwise. You can assume x is a list by islist(x) := 1. e.g. islist([a]) gives 1, islist(3.2) gives 0. See also: iseven, isinteger, isodd, isreal, isnumber, isfree, sgn. * isodd isodd(x) gives 1 if x is an odd integer, or 0 otherwise. You can assume x is odd by isodd(x) := 1. e.g. isodd(3) gives 1, isodd(2) gives 0. See also: iseven, isinteger, islist, isreal, isnumber, isfree, sgn. * isreal isreal(x) gives 1 if x is real, or 0 otherwise. You can assume x is real by isreal(x) := 1. e.g. isreal(2.2) gives 1, isreal(a) gives 0. See also: iseven, isodd, isinteger, islist, isnumber, isfree, sgn. * isnumber isnumber(x) gives 1 if x is a number, or 0 otherwise. You can assume x is a number by isnumber(x) := 1. e.g. isnumber(2.2) gives 1, isnumber(a) gives 0. See also: iseven, isodd, isinteger, islist, isreal, isfree, sgn. * last last represents the last output, as the built-in variable. last[1] the first element of the last output list. * left left(x===a) gives the left hand side of an equation. e.g. left(x+y===2) gives x+y. See also: righ. * length length([a]) gives the length of a list (the number of member in the list). e.g. length([a]) gives 1, length([a,b,c]) gives 3. See also: list. * lim lim(expr, x=x0) finds the limiting value of expr when x approaches x0. lim(expr, x=x0+zero) finds the right-sided limit as x approches x0 from the positive (+inf) direction (x -> x0+). lim(expr, x=x0-zero) finds the left-sided limit as x approches x0 from the negative (-inf) direction (x -> x0-). e.g. lim(sin(x)/x, x=0) gives 1. Note that the correct answers are only for the indeterminate forms: 0/0, inf/inf, 0*inf, 0^0, inf^0. See also: subs. * list list(f,x,xmin,xmax,dx) produces a list of f when x runs from xmin to xmax on step dx. e.g. list(x^2, x,1,3,1) gives [1,4,9]. See also: table, length. * ln ln(x) is the natural logarithmic function of x. Its base is e. It is the inverse to exp(x). Warming that if it has multi-values, the ln(x) only gives a principle value (P.V.) and other values are P.V.+2*k*pi*i (where k=0, 1, 2,..., -1, -2, ...). If x is complex number (x=A+B*i) then ln(x)=ln(abs(x))+i*atan2(A,B). See also: exp. * lnexpand The switch of the logarithmic expansion. lnexpand=on log expansion, e.g. ln(a*b) is expanded into ln(a)+ln(b). lnexpand=off disable log expansion, this is default. See also: expexpand, expand. * lowercase The swicth of the case conversion. lowercase=on converts the letters to lower-case letters, e.g. SIN(x) is converted to sin(x). lowercase=off disables the case convertion, this is default. It only effects the input. * mod mod(m,n) gives the remainder on division of m by n. See also: div. * n! n! gives the factorial of n. The same as fac(n). e.g. 3! gives 6. See also: fac. * num num(expr) gives the numerical value of expr. It converts all numbers to real form. e.g. num(pi) gives 3.1416. See also: numerical. * numerical The switch of numerical calculation. numerical=on numerical computation. numerical=off disable numerical computation, this is default. See also: num. * null null is a symbol used to indicate the absence of an expression or a result. When it appreas as an output expression, no output is printed. e.g. block(output=on, null). * off When the switch is set to off, it is inactive. e.g. numerical=off, output=off, expand=off. * on When the switch is set to on, it is active. e.g. numerical=on, expand=on, expexpand=on, lnexpand=on, lowercase=on, output=on. * openfile openfile('filename') opens the disk file 'filename' for writing. The filename is any MS-DOS file name. After doing something, the file must be closed by closefile('filename'). See also: closefile, include. * output The switch of the output format, e.g. output=basic, output=fortran, output=twodim, output=on, output=off. * pi pi=3.1416..., as the built-in constant, pi is converted to 3.1416... when the switch numerical=on. * plot plot(f,x,xmin,xmax,dx) generates a plot of f as a function of x from xmin to xmax with step dx. plot(f,x,xmin,xmax) plots a function with default step (xmax-xmin)* 0.05 (20 plot points). plot(f,x) plots a function with the default range (-6 to 6) and the default step (20 plot points). * prod prod(f,x,xmin,xmax,dx) evaluates the product of f when x runs from xmin to xmax with step dx. prod(f,x,xmin,xmax) with the default step dx = 1. * prolog prolog is the value of the switch output. It forces the output in the Prolog format. See also: output, basic, fortran. * random random(0) gives a uniformly distributed pseudorandom real in the range 0 to 1. random(n) gives a uniformly distributed pseudorandom integer in the range 0 to n. (n < 36000). e.g. random(0) gives 0.11111, random(5) gives 2. * ratio ratio(x) converts x to a rational number. e.g. ratio(0.2) gives 1/5. See also: float, num. * re re(x) gives the real part of the complex number x. e.g. re(1+2*i) gives 1. See also: im(x), abs(x), sgn(x). * repeat repeat(expr,test) repeats expr until test gives 1. e.g. x=1, repeat(x=x+1, x>5) gives x=6. See also: do. * right right(x===a) gives the right hand side of an equation. e.g. right(x+y === 3) gives 3. See also: left, solve. * round round(x) converts x to the rounded integer closest to x. e.g. round(2.4) gives 2, round(2.5) gives 3. See also: trunc. * sec sec(x) is the secant function of x. The angle x is measured in radians (multiply by degree to convert from degrees). sec(x)=1/cos(x). See also: asec, cos. * sech sech(x) is the hyperbolic secant function of x. sech(x)=1/cosh(x). See also: asech, cosh. * sgn sgn(x) is the sign function of x. Its value is 1, 0 or -1. / 1 if re(x) > 0, or re(x) = 0 and im(x) > 0; sgn(x) = 0 if x=0; \ -1 otherwise. You can assume x is positive or negative by sgn(x) := 1 or sgn(x) := -1. e.g. sgn(2) gives 1, sgn(1+i) gives 1. See also: abs, re, im. * sin sin(x) is the sine function of x. The angle x is measured in radians. (multiply by degree to convert from degrees). See also: asin, csc. * sinh sinh(x) is the hyperbolic sine function of x. sinh(x)=(exp(x)-exp(-z))/2. See also: asinh, acsch. * solve solve(x^2===0, x) solves a polynomial ordered up to 4. solve([expr1===expr2,expr3===expr4], [x,y]) solves systems of linear equations. It gives all symbolic solutions. e.g. solve(x^2+5*x+6===0, x), solve([x+y===3, x-y===1], [x,y]). See also: nsolve. * sqrt sqrt(x) is the square root function of x. It is the same as x^0.5. It only gives the principal value (P.V.) (sgn(sqrt(x)) >= 0). e.g. sqrt(4) gives 2, sqrt(2*i) gives 1+i. See also: ^. * subs subs(expr, x=x0) substitutes x in expr by x0. e.g. subs(x^2, x=a) gives a^2. See also: lim. * sum sum(f,x,xmin,xmax,dx) evaluates the sum of f when x runs from xmin to xmax with step dx. sum(f,x,xmin,xmax) with the default step dx = 1. e.g. sum(2^n,n,1,5,1.1), sum(x^n,n,1,5). See also: prod, list, table. * step step is the separator, the same as the comma (,). See also: from, to, , * system system(DOS) executes the operating system (DOS) command. e.g. system(dir). * table table(f,x,xmin,xmax,dx) produces a table of the function values when x runs from xmin to xmax with step dx. table(f,x,xmin,xmax) with the default step dx = 1. table(y[x],x,xmin,xmax,dx) transforms a list y into a table. e.g. table(x^2,x,1,4,1). See also: list, plot. * tan tan(x) is the tangent function of x. The angle x is measured in radians (multiply by degree to convert from degrees). See also: atan, cot. * tanh tanh(x) is the hyperbolic tangent function of x. See also: atanh, coth. * to to is the separator, the same as the comma (,). See also: from, step, ,. * trunc trunc(x) converts x to the truncated integer. e.g. trunc(2.9) gives 2. See also: round. * twodim towdim is a value of the switch output. It forces output in two dimension format. e.g. output=twodim. See also: output, off, on, basic, fortran, prolog. * undefined The built-in constant. It indicates that the value of the expression is undefined. e.g. the indeterminate forms: 0/0, inf/inf, 0*inf, 0^0. Users should try again by lim(f, x=x0). * zero The right-hand sided value at x=0, as the built-in constant. -zero is the left-sided limit from the negative direction. e.g. 1+zero is to approach to 1 from the positive (+infinity) direction (the right- hand sided value), and 1-zero is to approach to 1 from the negative (-infinity) direction (the left-hand sided value), i.e. limit as zero -> 0. e.g. exp(1/(0+zero)) gives inf, exp(1/(0-zero)) gives 0. + add or positive sign. - subtract or negative sign. * multiply. / divide. ^ power in BASIC, the same as ** in FORTRAN, e.g. 2^3 gives 8. ** power in FORTRAN, the same as ^ in BASIC. ! factorial, the same as fac(x), e.g. 3! or fac(3) gives 6. < less than. <= less than. > greater than. >= greater than. <> unequal a <> b gives 1 if a is not equal to b, 0 if a is equal to b, or left unevaluated otherwise. It only test two real numbers. e.g. 2 <> 3 gives 1, 2 <> 2 gives 0. := delayed assigment. = immediate assignment. == equal a==b gives 1 if a is equal to b, 0 if a is not equal to b, or left unevaluated otherwise. It can test two complex numbers or expressions. It gives 1 if a and b are the same expression or left unevaluated otherwise. e.g. 2==2 gives 1, a==a gives 1. === equation sign e.g. x^6===0 , comma The words from, to, step and comma are the same. # comment statement e.g. # this is demo. ------------------------------------------------------------------------- 14. References [1] Huang, W., Proceedings of the workshop on symbolic and numeric computation, Helsinki University, Finland, 1991, p 185-186. [2] Huang, W., Int. J. Math. Educ. Sci. Tech., 1992, 23(1), 160-165. [3] Microbit, IEEE Micro, 1992, 12(1), 76. [4] Huang, W., IEEE Micro, 1992, 12(3), 80. [5] Huang, W., Abs. Amer. Math. Soc., 1992, 13(3), 343-344. [6] Huang, W., Abs. Amer. Math. Soc., 1992, 13(5), 518. [7] Huang, W., Abs. Amer. Math. Soc., 1992, 13(6). [8] Huang, W., SIGSMALL/PC Notes, 1992, 18(1&2), 63-64. [9] Long, G., Australian PC World, 1992, June, 117-119.