[50] in mathematical software users group
symbmath 2.1
daemon@ATHENA.MIT.EDU (Weiguang Huang)
Thu Jul 30 02:49:39 1992
From: Weiguang Huang <huang@deakin.OZ.AU>
Date: Thu, 30 Jul 1992 16:36:36 +1000
To: books@prenhall.com, ecaxron@ariel.lerc.nasa.gov, icme-7@vm1.ulaval.ca,
sm21a.zip is availble now at rana.cc.deakin.oz.au, and will be on other ftp
sites soon.
------------------
SymbMath 2.1: A Symbolic Calculator with Learning
by Dr. Weiguang HUANG
5/6 Cara Road, Geelong, Vic. 3216, Australia
Phone: 61-52-443282, Email: huang@deakin.oz.au
SymbMath (an abbreviation for Symbolic Mathematics) is not
only a symbolic calculator but also an expert system that can solve
symbolical math problems. SymbMath will also perform exact numeric
computation. It can manipulate complicated formulas and return answers
in terms of symbols, formulas and exact numbers.
SymbMath is different from other software:
(1) It is able to learn from users. If users only input one
formula (e.g. one derivative) without writing any code,
it will automatically learn many problems (e.g.
derivatives, indefinite and definite integrals) related
to this formula.
(2) Users edit the library (e.g. integral tables) in "formula
format" by any text editor.
(3) It runs on small IBM PCs (8086) with 640 KB under MS-DOS.
Its capabilities include facilities to provide analytical and
numerical answers for:
o Differentiation: regular or higher order, partial or
total, mixed and even implicit differentiation,
one-sided derivatives.
o Integration: indefinite or definite integration,
multiple integration, infinity as a bound, parametric
integration, iterated integration, line or surface
integrals, discontinuous or implicit integration.
o Solution of equations: roots of a polynomial, systems of
algebraic or differential equations.
o Manipulation of expressions: simplification, factoring or
expansion, substitution, evaluation, built-in standard or
user-defined functions.
o Calculation: exact and floating-point numerical
computation of integer, rational, real and complex
numbers in the range from minus to plus infinity, even
with different units.
o Limits: real or complex limits, one-sided limits,
indeterminate forms.
o Complex: calculation, functions, derivatives, integration.
o Sum and product: finite or infinite, partial.
o Chemical calculation: molecular and atomic weights,
concentrations (all by entering the symbols of the
chemical elements).
o Chemical reactions: inorganic and organic.
o Other: series, lists, arrays, vectors, matrices, tables, etc.
Also included are:
o Pull-down and pop-up menus, resizable and colourable windows.
o Twin screen text editor with syntax check.
o Programming in integrated edit-interpret-debug environment.
o Reading and generating BASIC and FORTRAN codes.
o Displaying two-dimensional math notation.
o On-line help, and on-line manual.
o Interface with other software (e.g. with PlotData for graph).
It has three versions: Shareware, Student, and Advanced. The
shareware version lacking some advanced functions on input is available
from SIMTEL20 archives on anonymous FTP sites (wsmr-simtel20.army.mil as
/calculator/sm21a.zip, rana.cc.deakin.oz.au as /huang/sm21a.zip, etc) or
by e-mail from listserv@vm1.nodak.edu, listserv@ndsuvm1.bitnet. Please
refer to the README and SymbMath.DOC files in its package for detail.
----------------
Here are some problems that Maple and Mathematica cannot
solve, but SymbMath can do.
The following examples came from news on the sci.math.symbolic
newsgroup in 1991, and were run in Maple V, Mathematica 2.0, or
SymbMath 2.1.
************************ Example 1 ******************************
Maple:
> int(exp(-a * x^2), x=0..infinity);
infinity
/
| 2
| exp(- a x ) dx
|
/
0
# unevaluated.
# Declare 'a' non-negative:
> signum(a) := 1;
signum(a) := 1
# The same integral is now evaluated fully:
> int(exp(-a * x^2), x=0..infinity);
1/2
Pi
1/2 -----
1/2
a
SymbMath :
Input:
inte(exp(-a*x^2), x from 0 to inf)
assume(sqrt(a) > 0)
inte(exp(-a*x^2), x from 0 to inf)
Output:
1/2*a^(-0.5)*sqrt(pi)*erf(inf*sgn(sqrt(a)))
assumed
1/2*a^(-0.5)*sqrt(pi)
************************** Example 2 *********************************
Maple:
# Despite the fact that 'n' is declared non-negative...
> signum(n) := 1;
signum(n) := 1
# ...this simple proper definite integral (that any freshman calculus
# student can evaluate!) is left unevaluated:
> int(x^n, x=0..1);
1
/
| n
| x dx
|
/
0
SymbMath :
Input:
assume(n > -1)
inte(x^n, x from 0 to 1)
Output:
assumed
1/(1 + n)
************************** Example 3 *****************************
Maple:
> int(x^n, x=eps..1);
(n + 1)
1 eps
----- - ----------
n + 1 n + 1
# ...but the one-sided limit...
> limit(", eps=0, right);
(n + 1)
1 eps
limit ----- - ----------
eps -> 0+ n + 1 n + 1
# ...remains unevaluated...
> simplify(");
(n + 1)
- 1 + eps
limit - ----------------
eps -> 0+ n + 1
# ...no matter what we do!
> eval(");
(n + 1)
- 1 + eps
limit - ----------------
eps -> 0+ n + 1
SymbMath:
Input:
assume(n > -1)
inte(x^n, x from eps to 1)
subs(eps=0 to last)
Output:
assumed
1/(1 + n) - eps^(1 + n)/(1 + n)
1/(1 + n)
*************************** Example 4 ***************************
Maple:
> 0^n;
0
# Maple flags the error only when 'n' is replaced by the constant 0:
> 0^0;
Error, 0^0 is undefined
SymbMath:
Input:
assume(n > 0)
0^n
0^-n
0^0
Output:
assumed
0
discont
undefined
**************************** Example 5 ******************************
Maple:
> int(x^k, x);
(k + 1)
x
--------
k + 1
SymbMath:
Input:
inte(x^k*d(x))
subs(k=-1 to last)
Output:
constant + x^(1 + k)/(1 + k)
discont
**************************** Example 6 *****************************
Maple:
# The following limit is left unevaluated:
> limit(x^k/exp(x), x=infinity);
k
x
limit ------
x -> infinity exp(x)
# We might ask if Maple knows this result for specific (constant) values
# of the symbolic parameter 'k'. The answer is a QUALIFIED "yes".
# Maple knows the result for 'k' equal to 10^8...
> limit(x^(10^8)/exp(x), x=infinity);
0
# ...and Maple knows the result for 'k' equal to 10^9...
> limit(x^(10^9)/exp(x), x=infinity);
0
# ...BUT, Maple seems to FORGET the result when 'k' equals 10^10...
> limit(x^(10^10)/exp(x), x=infinity);
10000000000
x
limit ------------
x -> infinity exp(x)
SymbMath:
Input:
lim(x=inf, x^k/exp(x))
lim(x=inf, x^(10^10)/exp(x))
lim(x=inf, x^(10^10000)/exp(x))
Output:
0
0
0
****************************** Example 7 ****************************
Maple:
> int(x^m * exp (-b * x), x=0..infinity);
infinity
/
| m
| x exp(- b x) dx
|
/
0
# As expected, the integral remains unevaluated. Now declare the signs
# of the symbolic parameters:
> signum(b) := 1;
signum(b) := 1
> signum(m) := 1;
signum(m) := 1
# Upon attempting to compute the integral a second time...
> int(x^m * exp (-b * x), x=0..infinity);
infinity
/
| m
| x exp(- b x) dx
|
/
0
# ...THE INTEGRAL CONTINUES TO REMAIN UNEVALUATED, despite the fact that
# the signs of the parameters 'b' and 'm' were declared PRIOR to the second
# attempt at computation.
SymbMath:
Input:
inte(x^n*exp(-a*x), x from 0 to inf)
Output:
inte(x^n*exp(-a*x), x, 0, inf)
************************ Example 8 *********************************
Mathematica:
In[1]:= Integrate[1/x,{x,-1,1}]
Out[1]= -Log[-1]
Maple:
has the same problem.
SymbMath:
Input:
inte(1/x, x from -1 to 1)
inte(1/x, x from -1 to 2)
Output:
0
ln(2)
*************************** Example 9 *******************************
Maple:
has a problem for int(tan(x), x=0..pi).
SymbMath:
Input:
inte(tan(x), x from 0 to pi)
Output:
0
**************************** Example 10 ****************************
Mathematica:
cannot evaluate integral of sgn(x).
Maple:
# cannot evaluate inegral of signum(x) by int(). Help with a procedure:
# Load the procedure into Maple:
> read `pvint.txt`;
pvint := proc(f,x,a,b,s)
local i1,i2,eps;
signum(eps) := 1;
i1 := int(f,x = a .. s-eps);
i2 := int(f,x = s+eps .. b);
simplify(i1+i2);
limit(",eps = 0,right)
end
> pvint(signum(x), x, -1, 1, 0);
- eps 1
/ /
| |
limit | signum(x) dx + | signum(x) dx
eps -> 0+ | |
/ /
-1 eps
# Maple refuses to evaluate this P.V. integral:
> simplify(");
- eps 1
/ /
| |
limit | signum(x) dx + | signum(x) dx
eps -> 0+ | |
/ /
-1 eps
> eval(");
- eps 1
/ /
| |
limit | signum(x) dx + | signum(x) dx
eps -> 0+ | |
/ /
-1 eps
SymbMath:
Input:
inte(sgn(x), x from -1 to 1)
inte(sgn(x), x from -1 to 2)
Output:
0
1
************************* Example 11 ********************************
Implicit diff. gives 1+y'[x](1+1/y[x])==0; y'[x]==-y[x]/(y[x]+1).
Mathematica:
given this eq. as input to DSolve says that built-in
procedure can't solve it.
MACSYMA:
(c1) depends(y,x)$
(c2) ode2(diff(y,x) = -y/(y+1),y,x);
(d2) - log(y) - y = x + %c
(c3) method;
(d3) separable
SymbMath:
solve the differential equation by integration inte() or by dsolve().
Input:
d(y)/d(x)*(1+1/y) === -1
inte(last*d(x))
Expand=On
dsolve(d(y)/d(x) === -y/(y+1), y)
Output:
(1 + 1/y)*d(y)/d(x) === -1
y + ln(y*sgn(y)) === constant - x
Expand = On
-y - ln(y*sgn(y)) === constant + x
************************* Example 12 ********************************
Mathematica:
y'[x] = y[x]^(1/2)
y[0] = 0
DSolve could not handle it. (It rarely solves anything!!), and the
RungeKutta package only gave me the solution
y[x]=0
Obviously, there is another solution viz.
y[x] = (x/2)^2
SymbMath:
Input:
dsolve(d(y)/d(x) === sqrt(y), y)
(last/2)^2
Output:
2*sqrt(y) === constant + x
y === 1/4*(constant + x)^2
************************* Example 13 ********************************
Maple:
> sqrt(x*x);
x
Mathematica:
Sqrt[a^2] evaluates to Sqrt[a^2].
SymbMath:
Input:
sqrt(x^2)
assume(a > 0)
sqrt(a^2)
assume(b <0 )
sqrt(b^2)
Output:
x*sgn(x)
assumed
a
assumed
-b
********************** Example 14 **********************************
Maple and Mathematica cannot find the integrals of abs(x).
SymbMath:
Input:
inte(abs(x), x from -1 to 1)
inte(abs(x)^5*d(x))
Output:
1
constant + 1/6*abs(x)^6*sgn(x)
---------------------------------------------------------------------
The following problems are taken from Swokowski's Calculus
book. They cause Mathematica to fail because of a singularity in
the interior of the interval of integration:
Section 10.4 Problems 3, 12, 15, 16, 23, 29.
The comments "INTEGRAL IS DIVERGENT" and "Principal Value" come from
Macsyma. Mma gives no indication that anything is amiss.
************************ Problem3 *********************************
Mathematica:
In[4]:= Integrate[1/x^2,{x,-3,1}]
4
Out[4]= -(-) (* INTEGRAL IS DIVERGENT *)
3
SymbMath:
Input:
inte(1/x^2, x from -3 to 1)
Output:
inf
************************** Problem12 ***************************
Mathemtica:
In[6]:= Integrate[x^(-4/3),{x,-1,1}]
Out[6]= -6 (* INTEGRAL IS DIVERGENT *)
SymbMath:
Input:
inte(x^(-4/3), x from -1 to 1)
Output:
inf
**************************** Problem15 ************************
Mathematica:
In[7]:= Integrate[1/x,{x,-1,2}]
Out[7]= -Log[-1] + Log[2]
Maple:
has the same problem.
Macsyma:
(c7) integrate(1/x,x,-1,2);
Principal Value
(d7) log(2)
SymbMath:
Input:
inte(1/x, x from -1 to 2)
Output:
ln(2)
********************** Problem16 ************************************
Mathematica:
In[8]:= Integrate[1/(x^2-x-2),{x,0,4}]
-Log[-2] Log[2] Log[5]
Out[8]= -------- + ------ - ------
3 3 3
Macsyma:
(c8) integrate(1/(x^2-x-2),x,0,4);
Principal Value
log(5)
(d8) - ------
3
SymbMath:
Input:
inte(1/(x^2-x-2), x from 0 to 4)
Output:
-1/3*ln(2) + 1/3*ln(2/5)
***************************** Problem23 **********************
Mathematica:
In[10]:= Integrate[(1/x^2)Cos[1/x],{x,-1,2}]
1
Out[10]= Sin[-1] - Sin[-] (* INTEGRAL IS DIVERGENT *)
2
SymbMath:
Input:
y=1/x^2*cos(1/x)
inte(y, x from -1 to 0-zero) + inte(y, x from 0+zero to 2)
Output:
y = x^(-2)*cos(1/x)
sin(-1) - sin(1/2) + 2*sin(inf)
************************** Problem29 ********************************
Mathematica:
In[12]:= Integrate[1/(x-4)^2,{x,0,Infinity}]
1
Out[12]= -(-) (* INTEGRAL IS DIVERGENT *)
4
SymbMath:
Input:
y=1/(x-4)^2
inte(y, x from 0 to 4-zero) + inte(y, x from 4+zero to inf)
Output:
y = (-4 + x)^(-2)
inf
