MathHandbook.com
Computer
Algebra System with Learning
Dr. Weiguang HUANG
226 Anzac Pde, Kensington, Sydney, NSW
2033, Australia
Phone:
+61 (0) 413008019
E-mail: DrHuang@DrHuang.com
www.DrHuang.com
8/5/2015
Copyright (C) 1990-2015
Contents
PART 1
User's Guide............................................. 7
1. Introduction..................................................... 7
1.1. What is SymbMath......................................... 7
1.2.
Capabilities..................................................... 7
1.3. Shareware
and Advanced Versions.................... 8
1.4 A List of
Files in SymbMath................................. 8
2. Calculation......................................................... 10
2.1. Exact Calculation.............................................. 10
2.2.
Discontinuity and One-sided Value.................. 11
2.3. Undefined
and Indeterminate Form................... 11
3. Simplification..................................................... 12
3.1. Assuming Domain............................................. 13
3.2. Comparing and Testing Numbers...................... 14
4. Defining
Functions, Procedures and Rules 15
4.1. Defining
Functions......................................... 15
4.1.1. Defining
Conditional Functions................................................. 16
4.1.2. Defining
Case Functions.......................................................... 16
4.1.3. Defining
Piece-wise Functions................................................. 16
4.1.4. Defining
Recursion Functions.................................................. 17
4.1.5. Defining
Multi-Value Functions................................................. 17
4.2. Defining
Procedures......................................... 17
4.3. Defining
Rules................................................. 18
5. Limits............................................................... 18
5.1. One-sided
Limits.............................................. 18
5.2. Numeric
Limits: nlim()...................................... 20
6. Differentiation.................................................... 20
6.1. Fractional
Derivatives....................................... 21
6.2. Defining f
(x).................................................... 21
7. Integration......................................................... 22
7.1. Indefinite Integration.......................................... 22
7.2. Definite Integration............................................ 24
7.3. Numeric
Integration.......................................... 24
8. Solving
Equations.......................................... 25
8.1. Solving Algebraic Equations.............................. 25
8.2. Equation
Solver: solve()................................... 26
8.3. Polynomial
Solver: psolve().............................. 27
8.4. Numeric
Solver: nsolve().................................. 27
8.5. Solving
Differential Equations........................... 27
8.6.
Differential Solver: dsolve().............................. 28
9. Sums,
Products, and Series........................... 28
9.1. Partial
Sum...................................................... 29
9.2. Indefinite
Sum.................................................. 29
9.3. Series.............................................................. 29
9.4.
Polynomials..................................................... 30
10. Lists and Arrays, Vectors and Matrices.. 31
10.1. Lists............................................................. 31
10.1.1.
Entering Lists........................................................................ 31
10.1.2.
Accessing Lists.................................................................. 31
10.1.3.
Modifying Lists................................................................... 32
10.1.4.
Operating Lists.................................................................. 32
10.2. Arrays.......................................................... 34
10.2.1.
Entering Arrays..................................................................... 34
10.2.2.
Accessing Arrays.................................................................. 34
10.2.3.
Modifying Arrays................................................................... 34
10.2.4.
Operating Arrays.................................................................. 34
10.3. Vectors
and Matrices.................................. 35
11.
Statistics....................................................... 35
12. Tables
of Function Values......................... 35
13. Conversion.................................................... 36
13.1.
Converting to Numbers.................................. 36
13.2.
Converting to Lists......................................... 36
13.3.
Converting to Strings...................................... 37
13.4.
Converting to Table........................................ 37
14. Getting
Parts of Expression....................... 37
14.1. Getting
Type of Data...................................... 37
14.2. Getting
Operators........................................... 37
14.3. Getting
Operands........................................... 38
14.4. Getting
Coefficients........................................ 38
15. Database....................................................... 38
15.1. Phone
Number............................................... 39
15.2. Atomic
Weight................................................ 39
15.3. Chemical
Reaction......................................... 39
16. Learning
from User........................................ 39
16.1. Learning
Integrals from a Derivative............... 39
16.2. Learning
Complicated Integrals from a Simple Integral............................................................................... 41
16.3. Learning
Definite Integral from Indefinite Integral 42
16.4. Learning Complicated Derivatives from Simple
Derivative................................................................ 42
16.5. Learning
Integration from Algebra................... 42
16.6. Learning
Complicated Algebra from Simple Algebra............................................................................... 42
16.7. Learning
vs. Programming.............................. 43
PART 2
Programmer's Guide........................... 44
17. Programming
in SymbMath.......................... 44
17.1. Data
Types.................................................... 44
17.1.1. Numbers.............................................................................. 44
17.1.2.
Constants............................................................................. 45
17.1.3.
Variables.............................................................................. 46
17.1.4.
Patterns............................................................................... 47
17.1.5.
Functions, Procedures and Rules........................................... 47
17.1.5.1.
Standard Mathematical Functions....................................... 47
17.1.5.2.
Calculus Functions............................................................. 48
17.1.5.3. Test
Functions................................................................... 49
17.1.5.4.
Miscellaneous Functions..................................................... 51
17.1.5.5.
User-defined Functions.................................................... 52
17.1.5.6.
Procedures........................................................................ 52
17.1.5.7. Rules................................................................................. 52
17.1.6.
Equations............................................................................. 52
17.1.7.
Inequalities........................................................................... 53
17.1.8. Vectors
or Lists..................................................................... 53
17.1.9.
Matrices or Arrays................................................................ 53
17.1.10.
Strings................................................................................ 53
17.2.
Expressions.................................................. 54
17.2.1.
Operators............................................................................ 54
17.2.1.1.
Arithmetic Operators.......................................................... 55
17.2.1.2.
Relational Operators.......................................................... 55
17.2.1.3.
Logical Operators............................................................. 55
17.2.2.
Function Calls....................................................................... 56
17.3.
Statements.................................................. 56
17.3.1. Comment
Statements........................................................... 56
17.3.2.
Evaluation Statements.......................................................... 57
17.3.3.
Assignment Statements........................................................ 57
17.3.4.
Conditional........................................................................... 58
17.3.5. Loop.................................................................................... 58
17.3.6. Switch.................................................................................. 59
17.3.6.1.
Output Switch.................................................................... 59
17.3.6.2. Case
Switch....................................................................... 59
17.3.6.3.
Numeric Switch.................................................................. 59
17.3.6.4.
Expand Switch................................................................... 60
17.3.6.5.
ExpandExp Switch............................................................. 60
17.3.7. Read
and Write Statements................................................... 60
17.3.8. DOS
Command.................................................................... 61
17.3.9.
Sequence Statements........................................................... 61
17.4. Libraries
and Packages.................................. 61
17.4.1. Initial
Package init.sm............................................................ 63
17.4.2.
ExpandLn Package............................................................... 63
17.4.3.
Chemical Calculation Package............................................... 63
17.5. Interface
with Other Software................................................... 64
18. Graphics......................................................... 64
18.1. Drawing
Lines and Arcs.................................. 65
18.2. Plotting
f(x).................................................... 66
18.3. Plotting
Parametric Functions x(t) and y(t)...... 67
18.4. Plotting f(t) in Polar Coordinates................... 67
18.5. Plotting
Data.................................................. 68
18.6. Printing Graphics on Printer........................... 68
Part 3 Reference Guide........................................ 69
19. SymbMath
Environment: Windows and Menus................................................................................. 69
19.1. File Menu....................................................... 69
19.1.1. Open.................................................................................... 70
19.1.2. New..................................................................................... 70
19.1.3. Save
Input............................................................................ 70
19.1.4. Save
Output......................................................................... 70
19.1.5. DOS
Shell............................................................................ 70
19.1.6. Exit...................................................................................... 70
19.2. Input
Menu..................................................... 71
19.3. Run Menu...................................................... 71
19.4. Output
Menu.................................................. 71
19.5. Color
Menu.................................................... 71
19.5.1. Menu
Line............................................................................ 71
19.5.2. Input
Window........................................................................ 71
19.5.3. Input
Border......................................................................... 72
19.5.4. Output
Window..................................................................... 72
19.5.5. Output
Border....................................................................... 72
19.5.6. Status
Line........................................................................... 72
19.6. Help Menu..................................................... 72
19.7. Example
Menu............................................... 73
19.8. Keyword
Menu............................................... 73
19.9. Editor
and Edit Help Menu.............................. 73
19.9.1. Edit
Help Menu..................................................................... 73
19.9.1.1. Show
Help File................................................................... 74
19.9.1.2.
Cursor Movement Commands............................................ 74
19.9.1.3.
Insert and Delete Commands............................................. 74
19.9.1.4.
Search and Replace Commands......................................... 75
19.9.1.5. Block
Commands............................................................... 75
19.9.1.6.
Special Block Commands................................................... 77
19.9.1.9.
Miscellaneous Commands.................................................. 77
19.9.1.10.
Global Commands........................................................... 78
19.9.2. Edit
Commands.................................................................... 78
19.9.3. Copy and Paste..................................................................... 81
20. Inside
SymbMath........................................... 83
20.1. Internal
Structure........................................... 83
20.2. Internal
Format.............................................. 84
21. System Limits.................................................. 84
22. Keywords......................................................... 85
22.1.
Internal Functions....................................... 85
22.2.
External Functions...................................... 85
22.3.
Keywords in Alphabetical Order.................... 86
22.4.
Keywords in Topic Order............................. 87
22.5.
Glossary..................................................... 93
SymbMath (an abbreviation for Symbolic Mathematics) is a
symbolic calculator that can solve symbolic math problems.
SymbMath is a computer
algebra system that can perform exact, numeric, symbolic and graphic
computation. It manipulates complicated formulas and returns answers in terms
of symbols, formulas, exact numbers, tables and graph.
SymbMath is an expert system
that is able to learn from user's input. If the user only input one formula
without writing any code, it will automatically learn many problems related to
this formula (e.g. it learns many integrals involving an unknown function f(x)
from one derivative f’).
SymbMath is a symbolic,
numeric and graphics computing environment where you can set up, run and
document your calculation, draw your graph.
SymbMath uses external functions as if standard
functions since the external functions in library are auto-loaded.
SymbMath is a programming language in which you can define
conditional, case, piecewise, recursive, multi-value functions and procedures,
derivatives, integrals and rules.
SymbMath is database where
you can search your data.
It runs on IBM PCs (8086) with 400
KB free memory under MS-DOS.
It can provide analytical and numeric
answers for:
*
Differentiation: regular or higher order, partial or total, mixed and
implicit differentiation, one-sided derivatives.
* Integration:
indefinite or definite integration, multiple integration, infinity as a bound,
parametric or iterated implicit integration.
* Solution of
equations: roots of a polynomial, systems of algebraic or differential
equations.
* Manipulation
of expressions: simplification, factoring or expansion, substitution,
evaluation.
* Calculation:
exact and floating-point numeric computation of integer, rational, real and
complex numbers in the range from minus to plus infinity, even with different
units.
* Limits: real,
complex or one-sided limits, indeterminate forms.
* Complex:
calculation, functions, derivatives, integration.
* Sum and
product: partial, finite or infinite.
* Others: series,
lists, arrays, vectors, matrices, tables, etc.
Also included are:
*
External functions in library as if standard functions.
* Plot: functions, polar, parametric, data, and
list.
* Draw: lines,
arcs, ellipse, circles, ovals.
* Procedural, conditional, iteration, recursive,
functional, rule-based, logic, pattern-matching and graphic programming.
* Searching
database.
You should register with the author if you
use SymbMath.
Please read all * files before
running SymbMath. Please copy-and-past examples in the Help window to practise.
The printed documents (100+ pages) is available from author.
If you get the SymbMath on ZIP
format (e.g. sm32a.zip), you should unzip it with parameter -d by
pkunzip -d sm32a
c:\symbmath
If you get the SymbMath with the install
file, you should install it by install
On the MS-DOS prompt to run it, type SymbMath
SymbMath has two versions: Shareware
Version A, and Advanced Version C. The Shareware version lacks the solve(),
trig (except sin(x) and cos(x)), and hyperbolic functions, (lack 10% keywords).
You cannot input these lack functions in Shareware version.
------------------------------------------------------------------------------------------
Version Class Lacked functions
Shareware A solve(),
hyperbolic,
trig (except
sin(x), cos(x))
Advanced C
Libraries * . li
Manual printed
...........................................................................................................................
Upgrade same
...........................................................................................................................
Multiple copies >2
...........................................................................................................................
Site licence >10 copies
------------------------------------------------------------------------------------------
You must provide the photocopy of
your license or license number for upgrades.
If you send the author your payment
by cheque, money order or bank draft that must be drawn in Australia, you will
get the latest version. If you sign the license (see the LICENSE file) and send
it to the author, you will be a legal user for upgrades. If you write a paper
about this software on publication, you will get a free upgrade.
Its two versions (Shareware and
Advanced) are available from the author. The Shareware version is available
from my web sites.
The Advanced version is
copy-protected, so you must insert the original SymbMath disk into drive A or B
before you run SymbMath. By default, it is drive B. If you use drive A, please
copy (or rename) the DRIVE.A file to the SYMBMATH.DRI file, or you edit
drive(2) into drive(1) in the SYMBMATH.DRI file.
---------------------------------------------------------------------------------------------------------------
In directory
SymbMath:
SymbMath.exe executable SymbMath system.
SymbMath.ini initial file.
SymbMath.h* help
files, * is numbers.
SymbMath.dri indicate which drive the original
SymbMath disk is inserted into
init.sm initial SymbMath
program file.
*.sm packages (user
SymbMath program files).
prolog.err prolog error message file.
In directory
SymbMath\BGI:
*.bgi BGI graphics
drives.
*.chr stroked fonts.
In directory
SymbMath\library:
* the auto
loaded libraries (external functions).
In directory
SymbMath\keyword:
*.key the keyword files.
In directory
SymbMath\text:
SymbMath introduction of SymbMath.
readme the read-me file, this file should
be read first.
problem problems that other
software cannot solve, but SymbMath can do.
comment* comments on SymbMath.
statisti the download statistics at FTP
site of garbo.uwasa.fi.
shareware Shareware concept.
software software available from
the author.
update the latest updates in SymbMath.
------------------------------------------------------------------------
In the following examples, a line of
"IN: " means input, which you type in the Input window, then leave
the Input window by pressing <Esc>, finally run the program by the
command "Run"; while a line of "OUT: " means output. You
will see both input and output are displayed on two lines with beginning of
"IN: " and "OUT: " in the Output window. You should not
type the word "IN: ". Some outputs may be omitted on the examples.
#
is a comment statement.
You can split a line of command into
multi-lines of command by the comma ,. The comma without any blank space must be
the last character in the line.
Note that you should not be suprised
if some functions in the following examples are not working when their
libraries are not in the default directory or missing.
SymbMath gives the exact value of
calculation when the switch numeric := off (default), or the approximate value
of numeric calculation when the switch numeric := on or by num().
Mathematical functions are usually
not evaluated until by num() or by setting numeric := on.
SymbMath can manipulate units as
well as numbers, be used as a symbolic calculator, and do exact computation.
The range of real numbers is from -infinity to +infinity, e.g. ln(-inf),
exp(inf+pi*i), etc. SymbMath contains many algorithms for performing numeric
calculations. e.g. ln(-9), i^i, (-2.3)^(-3.2), 2^3^4^5^6^7^8^9, etc.
Note that SymbMath usually gives a
principle value if there are multi-values, but the solve() and root() give all
values.
Example:
Exact and numeric calculations of
1/2 + 1/3.
IN: 1/2+1/3 #
exact calculation
OUT: 5/6
IN: num(1/2+1/3) #
numeric calculation
OUT: 0.8333333333
Evaluate the value of the function
f(x) at x=x0 by f(x0).
Example:
Evaluate sin(x) when x=pi, x=180
degree, x=i.
IN: sin(pi), sin(180*degree)
OUT: 0, 0
IN: sin(i), num(sin(i))
OUT: sin(i),
1.175201 i
Example:
Set the units converter from the
minute to the second, then calculate numbers with different units.
IN: minute:=60*second
IN: v:=2*meter/second
IN: t:=2*minute
IN: d0:=10*meter
IN: v*t+d0
OUT: 250 meter
Evaluate the expression value by
subs(y, x = x0)
Example:
Evaluate z=x^2 when x=3 and y=4.
IN: z:=x^2 #
assign x^2 to z
IN: subs(z, x = 3) # evaluate z when x = 3
OUT: 9
IN: x:=4 #
assign 4 to x
IN: z #
evaluate z
OUT: 16
Note that after assignment of x by
x:=4, x should be cleared from assignment by clear(x) before differentiation
(or integration) of the function of x. Otherwise the x values still is 4 until
new values assigned. If evaluating z by the subs(), the variable x is automatically
cleared after evaluation, i.e. the variable x in subs() is local variable. The
operation by assignment is global while the operation by internal function is
local, but operation by external function is global. This rule also applies to
other operations.
The complex numbers, complex
infinity, and most math functions with the complex argument can be calculated.
Example .
IN: sign(1+i), sign(-1-i), i^2
OUT: 1, -1, -1
Example:
IN: exp(inf+pi*i)
OUT: -inf
IN: ln(last)
OUT: inf + pi*i
The built-in constants (e.g. inf,
zero, discont, undefined) can be used as numbers in calculation of expressions
or functions.
Some math functions are discontinuous at
x=x0, and only have one-sided function value. If the function f(x0) gives the
discont as its function value, you can get its one-sided function value by
f(x0-zero) or f(x0+zero).
Example:
IN: f(x_) := exp(1/x) # define function f(x)
IN: f(0)
OUT: discont # discontinuity at
x=0
IN: f(0-zero) #
left-sided value at x=0-
OUT: 0
IN: f(0+zero) #
right-sided value at x=0+
OUT: inf
If the function value is undefined, it may
be indeterminate form (e.g. 0/0, inf/inf), you can evaluate it by lim() (see
Chapter Limits).
SymbMath automatically simplifies the
output expression. You can further simplify it by using the built-in variable
last in a single line again and again until you are happy with the answer.
Expressions can be expanded by
expand(x)
expand := on
expandexp := on
Remember that the operation by
assignment is global while operation by function is local. So expand(x) only
expands the expression x, but the switch expand := on expands all expressions
between the switch expand := on and the switch expand := off. Second difference
betwen them is that the switch expand := on only expands a*(b+c) and (b+c)/p,
but does not expands the power (a+b)^2. The expandexp is exp expand.
Example:
IN: expand((a+b)^2+(b+c)*p)
OUT: a^2 + 2 a b +
b^2 + b p + c p
IN: expand := on
IN: (a+b)^2 + (b+c)*p
OUT: (a+b)^2 + b p +
c p
----------------------------------------------------------------------------------------
............... expand(x)
..........................................
(a+b)^2 to a^2+2*a*b+b^2
(a+b)^n to a^n+
...... +b^n n is positive integer
...............
expand(x) and expand := on ..........................
a*(b+c) to a*b
+ a*c
(b+c)/p to b/p
+ c/p
...............
expandexp := on .....................................
e^(a+b) to e^a * e^b
-----------------------------------------------------------------------------------------------
where a+b can be
many terms or a-b.
Expressions can be factorised by
factor(x)
e.g.
IN: factor(a^2 + 2*a*b + b^2)
OUT: (a + b)^2
Polynomials of order less than 5 can
be factorised by
factor(y, x)
Example:
IN: factor(x^2+5*x+6, x)
OUT: (2 + x) (3 + x)
Example:
Reduce sqrt(x^2).
IN: sqrt(x^2)
OUT: x*sign(x)
This output can be further
simplified if you know properties of x.
A first way is to evaluate x*sign(x)
by substituting sign(x) with 1 if x is positive.
IN: sqrt(x^2)
OUT: x*sign(x)
IN: subs(last, sign(x) = 1)
OUT: x
where a special
keyword last stands for the last output, e.g. here last is x*sign(x).
A second way is to assume x>0 before
evaluation. If you assume the variable x is positive or negative, the output
expression is simpler than that if you do not declare it.
IN: assume(x > 0, y <0) # assume x > 0, y < 0
OUT: assumed
IN: sqrt(x^2), sqrt(y^2), sqrt(z^2)
OUT: x*sign(x),
y*sign(y), z*sign(z)
IN: last # simplify last
output
OUT: x, -y,
z*sign(z)
In this way, all of x is affected
until the assume() is cleared by clear(). The first method is local
simplification, but the second method is global simplification.
By default, |x| < inf and all
variables are complex, except that variables in inequalities are real, as usual
only real numbers can be compared. e.g. x is complex in sin(x), but y is real
in y > 1.
Table 3.1 Assuming
--------------------------------------------------------------------------------------------------------
Assume Assignment Meaning
assume(x>y) x>y := 1 # assume x
> y
assume(x>=y) x>=y := 1 # assume x >= y
assume(x<y) x<y := 1 # assume x
< y
assume(x<=y) x<=y := 1 # assume x <= y
assume(x==y) x==y := 1 # assume x == y
assume(x<>y) x<>y := 1 # assume x <> y
iseven(b) :=
1 # assume b is even
isodd(b) :=
1 # assume b is odd
isinteger(b)
:= 1 # assume b is
integer
isratio(b)
:= 1 # assume b is ratio
isreal(b) :=
1 # assume b is real
iscomplex(b)
:= 1 # assume b is complex
isnumber(b)
:= 1 # assume b is number
islist(b) :=
1 # assume b is a list
isfree(y,x)
:= 1 # assume y is free of x
issame(a,b)
:= 1 # assume a is same as b
sign(b) := 1 # assume b is positive
complex
sign(b) :=
-1 # assume b is negative
complex
------------------------------------------------------------------------------------------------------------
The assume() can be cleared by
clear(). e.g. clear(x>y).
You can restrict the domain of a
variable by assuming the variable is even, odd, integer, real number, positive
or negative.
Example:
IN: isreal(b) := 1 # assume b is real
IN: sqrt(b^2)
OUT: abs(b)
Table 3.3 Simplification in different domains
----------------------------------------------------------------------------------
expression complex real x > 0
sqrt(x^2) x sign(x) |x| x
x*sign(x) x sign(x) |x| x
|x|*sign(x) |x| sign(x) x x
|x|/x |x|/x 1/sign(x) 1
x+inf x+inf inf
x-inf x-inf -inf
abs'(x) |x|/x 1/sign(x) 1
-------------------------------------------------------------------------------------
You can compare two numbers by relational
operators:
a > b
a < b
a <= b
a >= b
a <> b
a == b
Example:
IN: 2 > 1, 2 < 1
OUT: 1, 0
You also can compare two numbers,
even complex numbers z1 and z2 by
islarger(z1, z2)
isless(z1, z2)
issame(z1, z2)
Example:
compare 1+i and 1-i.
IN: islarger(1+i, 1-i) # is 1+i larger than 1-i ?
OUT: 1 # yes, 1+i >
1-i
You can compare square of a variable
a^2 > 0 if you know the property of the variable.
Example:
IN: assume(a > 0)
IN: a^2 > 0, 1/a > 0
OUT: 1, 1
You can test if x is even, odd,
integer, real, number or list by the functions:
iseven(x)
isodd(x)
isinteger(x)
isreal(x)
isnumber(x)
islist(x)
isfree(y,x)
islarger(a,b)
isless(a,b)
issame(a,b)
Example:
IN: iseven(2)
# is 2 even ?
OUT: 1 # yes
Note that comparison by the is*
functions return either 1 if it is true or 0 otherwise, but comparison by
relational operators gives 1 if it is true, 0 if it is fault, or left
unevaluated otherwise.
Anytime when you find yourself using the
same expression over and over, you should turn it into a function.
Anytime when you find yourself using
the same definition over and over, you should turn it into a library.
You can make your defined function
as if the built-in function, by saving your definition into disk file as a
library with the function name plus extension as the filename. e.g. saving the factoria function
as the factoria file (see Section Libraries and Packages).
You can define your own functions by
f(x_) := x^2
Here are some sample function
definitions:
f(x_) := cos(x + pi/3)
g(x_, y_) := x^2 - y^2
The argument in the definition
should be the pattern x_. Otherwise f() works only for a specific symbolic
value, e.g. x when defining f(x):=x^2. The pattern x_ should be only on the
left side of the assignment.
Once defined, functions can be used
in expressions or in other function definitions:
y := f(3.2)
z := g(4.1, -5.3)
Example:
Define a new
function for x^2, then evaluate it.
IN: g(x) := x^2
IN: g(2), g(a),
g(x)
OUT: g(2),
g(a), x^2 # work only for a symbolic value x
IN: f(x_) := x^2
IN: f(2), f(a)
OUT: 4, a^2 # work for any
value
You can define a conditional function by
the If() function:
f1(x_) := If(x>0 , 1)
f2(x_) := If(x>0 , x^2 , x)
or by inequalities:
f3(x_) := x>0
f4(x_) := (x>0) * x^2
+ (x<=0) * x
On the first definition by If(),
when f1() is called it gives 1 if x>0, or left unevaluated otherwise. On the
second definition by the If(), when f2() is called it gives x^2 if x>0, x if
x<=0, or left unevaluated otherwise. On the third definition by the
inequality, when f3() is called, it gives 1 for x>0, 0 for x<=0, or
x>0 for symbolic value of x. On the last definition, when f4() is called, it
is evaluated for any numeric or symbolic value of x.
You cannot differentiate nor integrate
the conditional function defined by If(), but you can do the conditional
functions defined by inequalities.
You can define a function evaluated
only for numbers by
f(x_) := If(isnumber(x) ,
x^2)
This definition is different from
the definition by f(x_) := x^2. On the latter, when f() is called, it gives
x^2, regardless whatever x is. On the former, when f() is called, it gives x^2
if x is a number, or left unevaluated otherwise.
Example:
evaluate to x^2 only
if x is number, by defining a conditional function.
IN: f(x_) := If(isnumber(x) , x^2)
IN: f(2), f(a)
OUT: 4, f(a)
IN: f(x_) := If(x>0 , x^2)
IN: f(2), f(-2), f(a)
OUT: 4, f(-2), f(a)
IN: f(x_) := If(x>0 , x^2 , x)
IN: f(2), f(-2), f(a)
OUT: 4, 2, f(a)
You can define the case function by
different pattern name. The case function is similar to the case statement in
BASIC language.
Example:
IN: f(x_) := If( x > 0 and x < 1 , 1)
IN: f(u_) := If( u > 1 and u < 2 , 2)
IN: f(v_) := If( v > 2 and v < 3 , 3)
IN: f(0.2), f(1.2), f(2.2)
OUT: 1, 2, 3
You can define a piece-wise function.
Example:
define
/ x if
x < 0
f(x) = 0 if
x = 0
\ x^2 if
x > 0
then evaluate f(-2),
f(0), f(3), f(a), f’, d(f(x), x=3).
IN: f(x_) := x*(x<0)+x^2*(x>0)
IN: f(-2), f(0), f(3), f(a)
OUT: -2, 0, 9, (a
< 0) a + (a > 0) a^2
IN: f’
OUT: (x < 0) + 2
x (x > 0)
IN: d(f(x), x=3)
OUT: 6
You can define a recursion function.
Example:
IN: factoria(1) := 1
IN: factoria(n_) := If(n > 1,
(n-1)*factoria(n-1))
You can define a function with the multi
function values.
Example:
IN: squreroot(x_) := [sqrt(x), -sqrt(x)]
IN: squreroot(4)
OUT: [2, -2]
You can define a function as a procedure by
f(x_) := block(command1,
command2, ..., commandN)
f(x_) := block(command1,
command2, ..., commandN, local(a))
By default, all variables within
procedure are global, except for variables declared by local(). The
multi-statement should be grouped by block(). The block() only outputs the
result of the last statement or the second last one as its value. The
multi-line must be terminated by a comma, (not by a comma and a blank space).
Local() must be the last one in block().
Example:
define a numeric
integration procedure ninte() and calculate integral of x^2 from x=1 to x=2 by
call ninte().
IN: ninte(y_,x_,a_,b_) := block( numeric:=on,
dd:=(b-a)/50,
aa:=a+dd,
bb:=b-dd,
y0:=subs(y, x = a),
yn:=subs(y, x = b),
(sum(y,x,aa,bb,dd)+(y0+yn)/2)*dd,
local(dd,aa,bb,y0,yn) )
IN: ninte(x^2, x, 1, 2)
You can define transform rules. Defining
rules is similar to defining functions. In defining functions, all arguments
must be simple variables, but in defining rules, the first argument can be a
complicated expression.
Example:
Define log rules.
IN: log(x_ * y_) := log(x) + log(y)
IN: log(x_ ^ n_) := n*log(x)
IN: log(a*b)
OUT: log(a) + log(b)
Example:
IN: sin(-x_) := -sin(x)
IN: sin(-a)
OUT: -sin(a)
Example:
Define derivatives (see Chapter
4.5.2 Defining f’).
IN: f'(x_) := sin(x)
IN: f'(t)
OUT: sin(t)
Example:
Define integrals (see Chapter 4.6.1
Indefinite Integration).
IN: integrate(f(x_),x_) := sin(x)
IN: integrate(f(t),t)
OUT: sin(t)
Example:
Define the trig simplification
rules.
IN: simplify(sin(x_)^2, x_) := 1/2*(1-cos(x))
IN: simplify(sin(x)^2,x)
OUT: 1/2 (1 -
cos(x))
Example:
Define Laplace transform rules.
IN: laplace(sin(t_), t_) := 1/(t^2+1)
IN: laplace(sin(s), s)
OUT: 1/(s^2 + 1)
You can finds real or complex limits, and
discontinuity or one-sided value.
First find the expression value by
subs(y, x = x0) or the function value by f(x0) when x = x0.
If the result is the discont (i.e.
discontinuity), then use the one-sided value x0+zero or x0-zero to try to find
the one-sided function or expression value.
For a function f(x), you can
evaluate the left- or right-sided function value, similar to evaluate the
normal function value:
f(x0-zero)
f(x0+zero)
For an expression y, you can
evaluate its one-sided expression value by
subs(y, x = x0-zero)
subs(y, x = x0+zero)
The discont (discontinuity) means
that the expression has a discontinuity and only has the one-sided value at
x=x0. You should use x0+zero or x0-zero to find the one-sided value. The value
of f(x0+zero) or f(x0-zero) is the right-sided or left-sided function value as
approaching x0 from positive (+inf) or negative (-inf) direction, respectively,
i.e. as x = x0+ or x = x0-.
If the result is undefined
(indeterminate forms, e.g. 0/0, inf/inf, 0*inf, and 0^0), then find its limit
by
lim(y, x = x0)
If the limit is discont, then you
can find a left- or right-sided limit when x approaches to x0 from positive
(+inf) or negative (-inf) direction at discontinuity by
lim(y, x = x0+zero)
lim(y, x = x0-zero)
Example:
Evaluate y=exp(1/x) at x=0, if the
result is discontinuity, find its left-sided and right-sided values (i.e. when
x approaches 0 from positive and negative directions).
IN: y:=exp(1/x)
IN: subs(y, x = 0)
OUT: discont # discontinuity at x=0
IN: subs(y, x = 0+zero), subs(y, x = 0-zero)
OUT: inf, 0
Example:
How to handle the following
one-sided values ?
Let f(x) = 1 when x
< 1, f(x) = 1 when x > 1 (and not defined at x = 1).
Let g(x) = 1 when x
< 1, g(x) = 1 when x > 1, and g(1) = 2.
Let h(x) = 1 when x
< 1, h(x) = 2 when x >= 1.
Let k(x) = 1 when x
< 1, k(x) = 2 when x > 1, and k(1) = 3.
Now ask SymbMath to
compute
(1) the limit as x
approaches 1,
(2) the limit as x
approaches 1 from the left, and
(3) the limit as x
approaches 1 from the right
for each of the
above piecewise defined functions.
# define functions
f(x_) := If(x<1 or x>1, 1)
f(1+zero):=1
f(1-zero):=1
g(x_) := If( x<1
or x>1, 1)
g(1):=2
g(1+zero):=1
g(1-zero):=1
h(x_) := If( x<1,
1, 2)
h(1+zero):=2
h(1-zero):=1
k(x_) := If( x<1,
1, If( x>1, 2))
k(1):=3
k(1+zero):=2
k(1-zero):=1
# evaluate functions
IN: f(1), g(1), h(1), k(1)
OUT: f(1), 2, 2, 3
IN: f(1+zero), g(1+zero), h(1+zero), k(1+zero)
# right-hand side
value at x=1+
OUT: 1, 1, 1, 1
IN: f(1-zero), g(1-zero), h(1-zero), k(1-zero)
# left-hand side
value at x=1-
OUT: 1, 1, 2, 2
Example:
Find limits of types
0/0 and inf/inf.
IN: p:=(x^2-4)/(2*x-4)
IN: subs(p, x = 2)
OUT: undefined
IN: lim(p, x = 2)
OUT: 2
IN: subs(p, x = inf)
OUT: undefined
IN: lim(p, x = inf)
OUT: inf
If symbolic limit falls, you should try
numeric limit by
nlim(y, x=x0)
e.g. nlim(sin(x)/x, x=0)
Differentiate an expression y with respect
to x by
d(y, x)
Differentiate a simple function f(x)
with respect to x by
d(f(x))
Differentiate y in the n order by
d(y, x, n)
Differentiate y at x = x0 by
d(y, x = x0)
Differentiate y at x = x0 in the n
order by
d(y, x = x0, n)
The order can be any number, e.g. 0.5
order is fractional derivative.
Example:
Differentiate sin(x) and x^(x^x).
IN: d(sin(x))
# it is the same as d(sin(x), x).
OUT: cos(x)
IN: d(x^(x^x), x)
OUT: x^(x^x) (x^(-1
+ x) + x^x ln(x) (1 + ln(x)))
If you differentiate f(x) by f’, x
must be a simple variable and f(x) must be unevaluated.
f'(x0) is the same as d(f(x0),x0),
but different from d(f(x), x=x0). f'(x0) first evaluates f(x0), then
differentiates the result of f(x0). But d(f(x), x=x0) first differentiates
f(x), then replace x with x0. Note that sin'(x^6) gives cos(x^6) as sin'(x^6)
is the same as d(sin(x^6), x^6). sin'(0) gives d(0,0) as sin(0) is evaluated to
0 before differentiation, you should use d(sin(x),x=0) which gives 1.
Example:
Differentiate the expression
f=sin(x^2+y^3)+cos(2*(x^2+y^3)) with respect to x, and with respect to both x
and y.
IN: f := sin(x^2+y^3)+cos(2*(x^2+y^3))
IN: d(f, x)
OUT: 2 x cos(x^2 +
y^3) - 4 x sin(2 (x^2 + y^3))
IN: d(d(f, x), y) # mixed derivative with x and y.
OUT: -6 x y^2
sin(x^2 + y^3) - 12 x y^2 cos(2 (x^2 + y^3))
Differentiate y at fractional order by
d(y, x , order)
d(y,x , -1.5)
Example:
IN: y:=d(sin(x), x, -1.5);
IN: d(y,
x,1.5)
The Davison-Essex
and the Riemann-Liouville definitions are different in the following aspect: in
the D-E formula, differentiation is performed first, then integration; in the
R-L formula it is the other way around. The D-E definition implemented, thus,
maps constants to zero, imitating integer order differentiation, while the R-L
definition does not. This property of the D-E definition makes it suitable to
work with initial value problems for fractional differential equations.
Defining derivatives is similar to defining
rules. You only need to define derivatives of a simple function, as SymbMath
automatically apply the chain rule to its complicated function.
Example:
IN: f(x_) := sin(x);
IN: f(x)
OUT: sin(x)
IN: f(x^6) # the same as d(f(x^6), x^6)
OUT: sin(x^6)
IN: d(f(x^6), x)
OUT: 6 x^5 sin(x^6)
You can find integrals of x^m*e^(x^n),
x^m*e^(-x^n), e^((a*x+b)^n), e^(-(a*x+b)^n), x^m*ln(x)^n, ln(a*x+b)^n, etc.,
(where m and n are any real number).
It is recommended that before you do
symbolic integration, you should simplify integrand, e.g. expand the integrand
by expand() and/or by setting the switch expand:=on and/or expandexp:=on.
If symbolic integration fails, you
can define a simple integral and/or derivative, (or adding integral into the int
library), then do integration again (see Chapter Learning From User).
Find the indefinite integrals of expr by
integrate(expr, x)
Find the double indefinite integrals
by
integrate(integrate(expr,
x), y)
Note that the arbitrary constant is
not represented.
Example:
Find integrals of 1/a, 1/b and 1/x,
knowing a >0, b is real.
IN: assume(a>0), isreal(b):=1
IN: integrate(1/a, a), integrate(1/b, b), integrate(1/x,
x)
OUT: ln(a), ln(|b|),
ln(x*sign(x))
Example:
Find indefinite integrals.
IN: integrate(sin(a*x+b), x) # integrands involving sin(x)
OUT: -cos(b + a x)/a
IN: integrate( sin(x)/x^2, x)
OUT: ci(x) -
sin(x)/x
IN: integrate( x*sin(x), x)
OUT: -x cos(x) +
sin(x)
IN: integrate(sin(x)*cos(x), x)
OUT: (1/2)*sin(x)^2
IN: integrate( e^(x^6), x) # integrands involving e^x
OUT: 1/6 ei(-5/6,
x^6)
IN: integrate( x^2*e^x, x)
OUT: ei(2, x)
IN: integrate( x*e^(-x), x)
OUT: -e^(-x) - x
e^(-x)
IN: integrate( e^x/sqrt(x), x)
OUT: ei(-0.5, x)
IN: integrate(x^1.5*exp(x), x)
OUT: ei(1.5, x)
IN: integrate(sin(x)*e^x, x) #
integrals involving sin(x) and e^x
OUT: 1/2 * (sin(x) -
cos(x)) * e^x
IN: integrate( x*ln(x), x) # integrands involving ln(x)
OUT: -1/4 x^2 + 1/2
x^2 ln(x)
IN: integrate( ln(x)^6, x)
OUT: li(6, x)
IN: integrate( ln(x)/sqrt(x), x)
OUT: -4 sqrt(x) + 2
sqrt(x) ln(x)
IN: integrate( ln(x)/sqrt(1 + x), x)
OUT: -4 sqrt(1 + x)
+ 2 sqrt(1 + x) ln(x) - 2 ln((-1 + sqrt(1 + x))/(1 + sqrt(1 + x)))
IN: integrate( 1/(a x + b), x) # integrands involving polynomials
OUT: ln((b + a x)
sign(b + a x))/a
IN: integrate( x/(x^2 + 5 x + 6), x)
OUT: 1/2 ln(|6 + 5 x
+ x^2|) - 5/2 ln(|(2 + x)/(3 + x)|)
IN: integrate( (x^3 + x)/(x^4 + 2 x^2 + 1), x)
OUT: 1/4 ln((1 + 2
x^2 + x^4) sign(1 + 2 x^2 + x^4))
Example:
Find the line integral.
IN: x:=2*t
IN: y:=3*t
IN: z:=5*t
IN: u:=x+y
IN: v:=x-y
IN: w:=x+y+z
IN: integrate(u*d(u,t)+v*d(v,t)+w*d(w,t), t)
OUT: 63 t^2
Example:
Integrate x^2*e^x, then expand it by
the mean of the packages "ExpandEi.sm" (expand ei(n,x)). The packages "ExpandGa.sm" (expand
gamma(n,x)) and "ExpandLi.sm" (expand li(n,x)) are similar one.
IN: integrate(x^2*e^x, x)
OUT: ei(2,x) # ei()
IN: readfile("ExpandEi.sm")
IN: ExpandEi(ei(2, x))
OUT: x^2 e^x - 2 x
e^x + 2 e^x # ei() is expanded
Defining integrals is similar to
defining rules.
Example:
IN: integrate(f(x_), x_) := sin(x)
IN: integrate(f(t), t)
OUT: sin(t)
Find definite integrals by external
functions
integrate(expr, x , xmin
, xmax)
integrate(expr, x , xmin
, singularity , xmax)
Example:
Find the definite integral of
y=exp(1-x) with respect to x taken from 0 to infinity.
IN: integrate(exp(1-x), x , 0 , inf)
OUT: e
Example:
do discontinuous
integration of 1/x^2 and 1/x^3 with discontinuity at x=0.
IN: integrate(1/x^2, x , -1 , 2) # singularity at x=0
OUT: inf
IN: integrate(1/x^3, x , -1 , 1) # singularity at x=0
OUT: 0
IN: integrate(sqrt((x-1)^2), x , 0 , 2) # singularity at x=1
OUT: 1
SymbMath usually detect singularity,
but sometime it cannot, in this case you must provide singularity.
Example:
IN: integrate(1/(x-1)^2, x , 0 , 1 , 2) # provide singularity at x=1
OUT: inf
Example:
complex integration.
IN: integrate(1/x, x , i , 2*i)
OUT: ln(2)
The external function
nint(y,
x , xmin , xmax)
does numeric
integration.
Example:
Compare numeric and
symbolic integrals of 4/(x^2+1) with respect to x taken from 0 to 1.
IN: nint(4/(x^2+1), x , 0 , 1)
OUT: 3.1415
IN: num(int(4/(x^2+1), x , 0 , 1))
OUT: 3.1416
The equations can be operated (e.g. +, -,
*, /, ^, expand(), d(), integrate()). The operation is done on both sides of
the equation, as by hand. You can find roots of a polynomial, algebraic
equations, systems of equations, differential and integral equations.
You can get the left side of the
equation by
left(left_side =
right_side)
or get the right
side by
right(left_side =
right_side)
You can assign equations to
variables.
Example:
IN: eq1:= x + y = 3
IN: eq2:= x - y = 1
IN: eq1+eq2
OUT: 2 x = 4
IN: last/2
OUT: x = 2
IN: eq1-eq2
OUT: 2 y = 2
IN: last/2
OUT: y = 1
Example:
Solve an equation sqrt(x+2*k) -
sqrt(x-k) = sqrt(k), then check the solution by substituting the root into the
equation.
IN: eq1 := sqrt(x + 2*k) - sqrt(x - k) = sqrt(k)
OUT: eq1 := sqrt(x +
2*k) - sqrt(x - k) = sqrt(k)
IN: eq1^2
OUT: ((2*k + x)^0.5
- ((-k) + x)^0.5)^2 = k
IN: expand(last)
OUT: 2*x + k +
(-2)*(2*k + x)^0.5*((-k) + x)^0.5 = k
IN: last-k-2*x
OUT: (-2)*(2*k +
x)^0.5*((-k) + x)^0.5 = (-2)*x
IN: last/(-2)
OUT: (2*k +
x)^0.5*((-k) + x)^0.5 = x
IN: last^2
OUT: (2*k + x)*((-k)
+ x) = x^2
IN: expand(last)
OUT: (-2)*k^2 + k*x
+ x^2 = x^2
IN: last-x^2+2*k^2
OUT: k*x = 2*k^2
IN: last/k
OUT: x = 2*k
IN: subs(eq1, x = right(last))
OUT: k^0.5 = k^0.5
You can solve algebraic equations
step by step, as above. This method is useful in teaching, e.g. showing
students how to solve equations.
The solve() functions
solve(expr1 = expr2, x)
solve([expr1 = expr2,
expr3 = expr4], [x, y])
solve a polynomial
and systems of linear equations on one step. It is recommended to set the
switch expand:=on when solve the complicated equations. All of the real and
complex roots of the equation will be found by solve(). The function solve()
outputs a list of roots when there are multi-roots. You can get one of roots
from the list, (see Chapter 4.9 Arrays, Lists, Vectors and Matrices).
Example:
Solve a+b*x+x^2 = 0 for x, save the root
to x.
IN: solve(a+b*x+x^2 = 0, x) # solve or re-arrange the equation for x
OUT: x = [-b/2 +
sqrt((b/2)^2 - a), -b/2 - sqrt((b/2)^2 -
a)]
IN: x := right(last) # assign two roots to x
OUT: x := [-b/2 +
sqrt((b/2)^2 - a), -b/2 - sqrt((b/2)^2 -
a)]
IN: x[1] #
the first root
OUT: -b/2 +
sqrt((b/2)^2 - a)
IN: x[2] #
the second root
OUT: -b/2 -
sqrt((b/2)^2 - a)
Example:
Solve x^3 + x^2 + x + 5 = 2*x + 6.
IN: num(solve(x^3+x^2+x+5 = 2*x+6, x))
OUT: x = [1, -1, -1]
The function solve() not only solves
for a simple variable x but also solves for an unknown function, e.g. ln(x).
Example:
Solve the equation for ln(x).
IN: solve(ln(x)^2+5*ln(x) = -6, ln(x))
OUT: ln(x) = [-2,
-3]
IN: exp(last)
OUT: x = [exp(-2),
exp(-3)]
Example:
Rearrange the equations.
IN: eq := [x+y = 3+a+b, x-y = 1+a-b] # assign equations to eq
IN: solve(eq, [x,y]) # rearrange eq for x and y
OUT: [x = -1/2*(-4 -
2 a), y = -1/2*(-2 - 2 b)]
IN: solve(eq, [a,b]) # rearrange eq for a and b
OUT: [a = -1/2*(4 -
2 x), b = -1/2*(2 - 2 y)]
IN: solve(eq, [a,y]) # rearrange eq for a and y
OUT: [b = -1/2*(2 -
2 y), x = -1/2*(-4 - 2 a)]
IN: solve(eq, [x,b]) # rearrange eq for x and b
OUT: [a = 1/2*(-4 +
2 x), y = 1/2*(2 + 2 b)]
The external function
psolve(f(x), x)
solves f(x)=0 for x.
It is similar to solve(), but it only can solve polynomial with order < 3.
e.g.
IN: psolve(x^2+5*x+6, x)
OUT: [-2, -3]
The external functions
nsolve(f(x) = x, x)
nsolve(f(x) = x, x,x0)
numerically solves
an algebraic equation with an initial value x0. By default x0=1. nsolve() only
gives one solution near x0, omitting other solutions.
Example:
IN: nsolve( cos(x) = x, x)
OUT: x =
0.73911289091
IN: nsolve( sin(x) = 0, x,0) # similar to asin( sin(x)=0 )
OUT: x = 0 #
only gives one solution near x0=0
IN: nsolve( sin(x) = 0, x,3)
OUT: x = 3.14 # only
gives one solution near x0=3
You can solve the differential equations:
y' = f(x)
by integrating the
equation.
y' is the same as d(y(x),x).
Example:
solve y'=sin(x) by integration.
IN: integrate( y' = sin(x), x)
OUT: y(x) = constant
- cos(x)
The external function
dsolve(y'(x) = f(x,y),
y(x), x)
can solve the first
order variables separable and linear differential equations
y'(x) = h(x)
y'(x) = f(y(x))
y'(x) = f(y(x))*x
y'(x) = g(x)*y(x)
y'(x) = g(x)*y(x)+h(x)
on one step. Notice
that y'(x) must be alone on the left hand side of the equation. It is
recommended to set the switch expand:=on when solving the complicated
differential equations.
Example:
Solve y'(x) = sin(x) by dsolve().
IN: dsolve( y'(x) = sin(x), y(x), x)
OUT: y(x) = constant
- cos(x)
Example:
Solve differential equations by
dsolve(). If the result is a polynomial, then rearrange the equation by
solve().
IN: dsolve(y'(x) = x/(2+y(x)), y(x), x)
OUT: 2*y(x) +
1/2*y(x)^2 = constant + x^2
IN: solve(last, y(x))
OUT: y(x) = [-2 +
sqrt(4 - 2*(-constant - x^2)),
-2 - sqrt(4 - 2*(-constant - x^2))]
Example:
Solve differential equations by
dsolve().
IN: dsolve(y'(x) = x*exp(y(x)), y(x), x)
OUT: -e^(-y(x)) =
constant + x^2
IN: dsolve(y'(x) = y(x)^2+5*y(x)+6, y(x), x)
OUT: ln((4 + 2
y(x))/(6 + 2 y(x))) = constant + x
IN: dsolve(y'(x) = y(x)/x, y(x), x)
OUT: y(x) = constant
x sign(x)
IN: dsolve(y'(x) = x + y(x), y(x), x)
OUT: y(x) = -1 - x +
constant*e^x
You can compute partial, finite or infinite
sums and products. Sums and products can be differentiated and integrated. You
construct functions like Taylor polynomials or finite Fourier series. The
procedure is the same for sums as products so all examples will be restricted
to sums. The general formats for these
functions are:
sum(y,x)
sum(y)
sum(expr, x , xmin ,
xmax)
sum(expr, x , xmin ,
xmax , dx)
prod(expr, x , xmin ,
xmax)
prod(expr, x , xmin ,
xmax , dx)
The expression expr is evaluated at
xmin, xmin+dx, ... up to the last entry
in the series not greater than xmax, and the resulting values are added or
multiplied. The part "step dx"
is optional and defaults to 1. The
values of xmin, xmax and dx can be any real number.
Here are some examples:
sum(j, j , 1 , 10,1)
for 1 + 2 + .. + 10.
sum(3^j, j , 0 , 10 , 2)
for 1 + 3^2 + ... +
3^10.
Here are some sample Taylor
polynomials:
sum(x^j/j!, j , 0 , n)
for exp(x).
sum((-1)^j*x^(2*j+1)/(2*j+1)!, j , 0
, n)
for sin(x) of degree
2*n+2.
Remember, the 3 keywords (from, to
and step) can be replaced by the comma ,.
The function
sum(f(x),x, a,b)
finds the partial
sum (symbolic sum).
Example:
Find the sum of 1^2
+ 2^2 ... + n^2.
IN: sum(x^2, x, 0,n)
OUT: 1/6 *n*(1 + n) *(1
+ 2* n)
The function
sum(f(x), x)
finds the indefinite
sum, i.e. inverse of the difference F(x+1)-F(x).
Example:
IN: sum(x)
OUT: x^2/2-x/2
This is different
from the partial sum(x,x,0,n).
The external functions
series(f(x), x)
series(f(x), x, order)
to find the Taylor
series at x=0. The argument (order) is optional and defaults to 5.
Example:
Find the power series expansion for
cos(x) at x=0.
IN: series(cos(x), x)
OUT: 1 - 1/2 x^2 +
1/24 x^4
The series expansion of f(x) is
useful for numeric calculation of f(x). If you can provide derivative of any
function of f(x) and f(0), even though f(x) is unknown, you may be able to
calculate the function value at any x, by series expansion. Accuracy of calculation
depends on the order of series expansion. Higher order, more accuracy, but
longer calculation time.
Example:
calculate f(1), knowing f’=-sin(x)
and f(0)=1, where f(x) is unknown.
IN: f'(x_) := -sin(x)
IN: f(0) := 1
IN: f(x_) := eval(series(f(x), x)) # must eval()
OUT: f(x_) := 1 -
1/2 x^2 + 1/24 x^4
IN: f(1)
OUT: 13/24
Polynomials are automatically sorted in
order from low to high.
You can pick up one of coefficient
of x in polynomials by
coef(poly, x^n)
e.g.
IN: coef(x^2+5*x+6, x)
OUT: 5
Note that you cannot pick up the
coefficient of x^0 by coef(y,x^0).
You can pick up one of coefficient
of x in polynomials with order < 5 by
coef(poly, x,n)
e.g.
IN: coef(x^2+5*x+6, x,0)
OUT: 6
You can pick up all of coefficients
of x in polynomials with order < 5 by
coefall(poly, x)
e.g.
IN: coefall(x^2+5*x+6, x)
OUT: [6, 5, 1] # 6 + 5*x + x^2
IN: coefall(a*x^2+b*x+c, x)
OUT: [c, b, a] # symbolic values of
coefficients
You can pick up the highest order of
x in polynomials with order < 5 by
order(poly, x)
e.g.
IN: order(x^2+5*x+6, x)
OUT: 2
You can factor polynomials in order
< 5 with respect to x by
factor(poly, x)
e.g.
IN: factor(x^2+5*x+6, x)
OUT: (2 + x) (3 + x)
You can construct lists and arrays of
arbitrary length, and the entries in the lists and arrays can be of any type of
value whatsoever: constants, expressions with undefined variables, or
equations.
A vector or matrix can be
represented by a list or array. In a matrix, the number of elements in each row
should be the same, e.g. [[a11, a12], [a21, a22]].
You can define a list by putting its
elements between two square brackets. e.g. [1,2,3]
You can define lists another way,
with the command:
[ list(f(x), x from xmin
to xmax step dx) ]
This is similar to the sum
command, but the result is a list:
[f(xmin), f(xmin+dx),
..., f(xmin+x*dx), ...]
which continues
until the last value of xmin + x*dx
<= xmax.
You also can assign the list to a
variable, which variable name become the list name:
a := [1,2,3] # define the list of a
b := [f(2), g(1),
h(1)] # assumes f,g,h defined
c := [[1,2],3,[4,5]] # define the list of c
Lists are another kind of value in
SymbMath, and they can be assigned to variables just like simple values. (Since
variables in SymbMath language are untyped, you can assign any value to any
variable.).
A function can have a list for its
value:
f(x_) := [sqrt(x),
-sqrt(x)]
e.g.
IN: squreroot(x_) := [sqrt(x), -sqrt(x)]
IN: squreroot(4)
OUT: [2, -2]
A function can have a list for its
argument:
abs([-1,2])
Try
a := [ list(j^2, j from
0 to 10 step 1) ]
f(x_) := [ list(x^j, j
from 0 to 6 step 1) ]
b := f(-2)
You can find the value of the j-th member
in a list by
member([a,b], j)
The first member of
a list is always member(x, 1).
If you have assigned a list to a variable
x, you can access the j-th element by the list index x[j]. The first element of
x is always x[1]. If the x[j] itself is a list, then its j-th
element is accessed by repeating the similar step.But you can not use the list
index unless the list is already assigned to x.
e.g.
IN: x := [[1,2],3,[4,5]] # define the x list
IN: x[1], x[2] #
take its first and 2nd element
OUT: [1, 2], 3
IN: x #
access the entire list of x
OUT: [[1, 2], 3,
[4,5]]
IN: member(x, 2) #
same as x[2]
OUT: 3
An entire sub-list of a list x can be accessed with the command x[j], which
is the list:
[x[j], x[j+1], ... ]
The subs() replaces the value of the
element in the list, as in the variables. e.g.
IN: subs([a,b,c], a = a0)
OUT: [a0, b, c]
Note that you cannot modify lists by
assignment.
Lists can be added, subtracted, multiplied,
and divided by other lists or by constants.
When two lists are combined, they are combined term-by-term, and the combination
stops when the shortest list is exhausted.
When a scalar is combined with a list, it is combined with each element
of the list. Try:
a := [1,2,3]
b := [4,5,6]
a + b
a / b
3 * a
b - 4
Example 4.9.2.4.1.
Two lists are added.
IN: [a1,a2,a3] + [b1,b2,b3]
OUT: [a1 + b1, a2 +
b2, a3 + b3]
IN: last[1]
OUT: a1 + b1
If L is a list, then f(L) results in a list of the values, even
though f() is the differentiation or integration function (d() or integrate()).
IN: sqrt([a, b, c])
OUT: [sqrt(a),
sqrt(b), sqrt(c)]
IN: d([x, x^2, x^3], x)
OUT: [1, 2*x, 3*x^2]
If you use a list as the value of a
variable in a function, SymbMath will try to use the list in the calculation.
You can sum all the elements in a
list x by
listsum(x)
Example:
IN: listsum([a,b,c]^2)
OUT: a^2 + b^2 + c^2
This function takes the sum of the
squares of all the elements in the list x.
You can do other statistical
operations (see Section 4.10. Statistics) on the list, or plot the list of
numeric data (see Section 5. Plot).
You can find the length of a list
(the number of elements in a list) with:
length(a)
You can define an array of a by assigning
its element value into its index:
a[1]:=1
a[2]:=4
or you can define arrays
another way, with the command:
do(a[x]:=f(x), x from
xmin to xmax step dx)
e.g.
do(a[j] := 2*j, j from 1 to 2)
You can define 2-dimentional array
by
a[1,1]:=11
a[1,2]:=12
a[2,1]:=21
a[2,2]:=22
or
do(do(a[j,k]:=j+k, j,jmin,jmax,dj),
k,kmin,kmax,dk)
After defining an array of a, you can
access one of its element by its index
IN: a[1]
OUT: 1
You also can list out all of its
elements by
list(a[j], j,1,2,1)
e.g.
IN: do(a[j]:=2*j, j,1,2,1)
IN: list(a[j], j,1,2)
OUT: 1, 4
You can modify an array by assigning new
value into its index
IN: a[1]:=2
e.g.
after defining 2 arrays a and b,
find their dot time, a .* b.
IN: a[1]:=1, a[2]:=2 # define array a
IN: b[1]:=11, b[2]:=12 # define array b
IN: p:=0
IN: do(p:=p + a[j]*b[j], j,1,2,1) # a .* b
You can uses arrays or lists to represent
vectors, and lists of lists to represent matrices.
Vectors and matrices can be operated
by "+" and "-" with vectors and matrixes, by "*"
and "/" with a scalar, and by d() and integrate(). These operations
are on each element, as in lists.
You can use lists as vectors, adding
them and multiplying them by scalars. For example, the dot product of two
vectors of a and b is:
sum(a[j]*b[j], j from 1
to jmax)
You can even make this into a
function:
dottime(x_, y_) :=
listsum(x*y)
e.g.
represent the dot product of two
vectors by arrays
IN: a[1]:=1, a[2]:=2 # define array a
IN: b[1]:=11, b[2]:=12 # define array b
IN: p:=0
IN: do(p:=p + a[j]*b[j], j,1,2,1) # a .* b
represent the dot product of two
vectors by lists
IN: dottime([1,2], [11,12]) # by lists in function dottime()
How about the cross product:
cross(a,b) =
[a[2]*b[3]-b[2]*a[3],a[3]*b[1]-b[3]*a[1],a[1]*b[2]-b[1]*a[2]]
Some statistical functions are:
average(x), max(x),
min(x), listsum(x), length(x)
A list of numbers can be calculation
on statistics.
Example:
IN: p := [1, 2, 3]
IN: average(p), max(p), min(p), length(p)
OUT: 2, 3, 1, 3
Not only a list of number but also a
list of symbolic data can be operated by some statistic functions to show how
to do the statistic operation.
IN: p := [a, b, c]
IN: average(p)
OUT: 1/3*(a + b + c)
IN: listsum(p)
OUT: a + b + c
IN: length(p)
OUT: 3
If you want to look at a table of values
for a formula, you can use the table command:
table(f(x), x)
table(f(x),
x from xmin to xmax)
table(f(x), x from xmin
to xmax step dx)
It causes a table of values for f(x)
to be displayed with x=xmin, xmin+dx, ..., xmax. If xmin, xmax, and step omit, then xmin=-5,
xmax=5, and dx=1 for default. You can specify a function to be in table(),
Example:
Make a table of x^2.
IN: table(x^2, x)
OUT:
-5, 25
-4, 16
-3, 9
-2, 4
: :
: :
Its output can be written into a
disk file for interfacing with other software (e.g. the numeric computation
software).
Different types of data may be converted
each other.
The complex number is converted to the real
number by
re(z), im(z), abs(z),
arg(z), sign(z)
The real number is converted to the
integer number by
trunc(x)
round(x)
The real number is converted to the
rational number by
ratio(x)
The rational number is converted to
the real number by
num(x)
numeric:=on
The rational number is converted to
the integer number by
nume(x)
deno(x)
The string is converted to the real
number if possible, by
number("123")
You can convert sum to a list of terms by
term(a+b)
IN: term(a+b)
OUT: [a, b]
You can convert product to a list of
multipliers by
mult(a*b)
IN: mult(a*b)
OUT: [a, b]
You can convert an array x to a list
by
[ list(x[j], j,1,jmax,1)
]
You can convert numbers to strings by
string(123)
IN: string(123)
OUT: "123"
A list of real numbers can be converted to
a table by
table()
Example:
IN: x := [5,4,3,2,1]
IN: table(x[j], j from 1 to 4 step 1)
OUT:
1, 5
2, 4
3, 3
4, 2
You can get type of data by
type(x)
IN: type(2)
OUT:
"integer"
You also can get operators by
type(x)
IN: type(a>b)
OUT:
">"
IN: type(sin(x))
OUT:
"sin()"
The functions
left(x=a), left(a >
b)
right(x=a), right(a >
b)
pick up the left-
and right- side of the equation and inequality.
IN: left(a>b), right(a>b)
OUT: a, b
You can get the j-th term of sum by
member(term(a+b), j)
IN: member(term(a+b), 1)
OUT: a
You can get the arguments of a
function by
argue(f(x))
IN: argue(sin(x))
OUT: x
A coefficient of x^n in an expression can
be picked up by
coef(p, x^n)
e.g.
IN: coef(a + b*x + c*x^2 + d*x^3, x)
OUT: b
You can get a coefficient of x^n
(where 0<= n < 4) in polynomials ordered up to 4 by
coef(poly, x,n)
(see Chapter
Polynomials for detail).
After you create a database file as a
library (external function), you can search your data by finding its function
value.
If you have created the database file
"phoneNo" as follow:
-------------------------------------
phoneno("huang")
:= "6974643"
phoneno("john") := "12345"
--------------------------------------
You can find out phone No. of
someone from the phone No. database file "phoneNo" by the external
function
phoneno("name")
Example:
find out phone No. of huang.
IN: phoneno("huang")
OUT: 6974643
You can search atomic weight of chemical
element from the database file "atom_wei".
e.g.
What is atomic weight of chemical
element H ?
IN: atom_wei(H)
OUT: 1
You can predict products for chemical
reactions from the database file "react".
Example 4.14.2.
What are the products when HCl +
NaOH react ?
IN: react(HCl+NaOH)
OUT: H2O + NaCl
One of the most important feature of
SymbMath is its ability to deduce and expand its knowledge. If you provide it
with the necessary facts, SymbMath can solve many problems which were unable to
be solved before. The followings are several ways in which SymbMath is able to
learn from your input.
Finding derivatives is much easier than
finding integrals. Therefore, you can find the integrals of a function from the
derivative of that function.
If you provide the derivative of a
known or unknown function, SymbMath can deduce the indefinite and definite
integrals of that function. If the function is not a simple function, you only
need to provide the derivative of its simple function. For example, you want to
evaluate the integral of f(a*x+b), you only need to provide f’.
If you know a derivative of an
function f(x) (where f(x) is a known or unknown function), SymbMath can learn
the integrals of that function from its derivative.
Example:
check SymbMath whether or not it had
already known integral of f(x)
IN: integrate(f(x), x)
OUT: integrate(f(x),
x)
IN: integrate(f(x), x, 1, 2)
OUT: integrate(f(x),
x, 1, 2)
As the output displayed only what
was typed in the input without any computed results, imply that SymbMath has no
knowledge of the indefinite and definite integrals of the functions in
question. Now you teach SymbMath the derivative of f(x) on the first line, and
then run the program again.
IN: f'(x_) := exp(x)/x
IN: integrate(f(x), x)
OUT: x*f(x) - e^x
IN: integrate(f(x), x, 1, 2)
OUT: e - f(1) +
2*f(2) - e^2
As demonstrated, you only supplied
the derivative of the function, and in exchange SymbMath logically deduced its
integral.
Another example is
IN: f'(x_) := 1/sqrt(1-x^2)
IN: integrate(f(x), x)
OUT: sqrt(1 - x^2) +
x*f(x)
IN: integrate(k*f(a*x+b), x)
OUT: k*(sqrt(1 - (b
+ a*x)^2) + (b + a*x)*f(b + a*x))/a
IN: integrate(x*f(a*x^2+b), x)
OUT: sqrt(1-(a*x^2 +
b)^2) + (a*x^2 + b)*f(a*x^2 + b)
The derivative of the function that
you supplied can be another derivative or integral.
Example:
IN: f'(x_) := eval(integrate(cos(x),x))
OUT: f'(x_) :=
sin(x)
IN: integrate(f(x), x)
OUT: -sin(x)
IN: integrate(f(a*x + b), x)
OUT: -sin(b + a*x)/a
IN: integrate(x*f(x), x)
OUT: -cos(x) -
x*sin(x)
IN: integrate(x^1.5*f(x), x)
OUT: 1.5*integrate(sqrt(x)*sin(x),
x) - x^1.5*sin(x)
IN: integrate(x^2*f(x), x)
OUT: -2*x*cos(x) +
2*sin(x) - x^2*sin(x)
IN: integrate(x*f(x^2), x)
OUT: -sin(x^2)
IN: integrate(x^3*f(x^2), x)
OUT: -0.5*cos(x^2) -
0.5*x^2*sin(x^2)
IN: integrate(f(x)/(x^1.5), x)
OUT: -2/sqrt(x)*f(x)
+ 2*integrate(sin(x)/sqrt(x), x)
IN: integrate(f(x)/(x^2), x)
OUT: -f(x)/x + si(x)
You supply a simple indefinite integral,
and in return, SymbMath will perform the related complicated integrals.
Example:
Check whether SymbMath has already
known the following integrals or not.
IN: integrate(f(x), x)
OUT: integrate(f(x),
x)
IN: integrate((2*f(x)+x), x)
OUT: integrate((2*f(x)+x),
x)
IN: integrate(integrate(f(x)+y), x), y)
OUT: integrate(integrate(f(x)+y),
x), y)
Supply, like in the previous
examples, the information: integral of f(x) is f(x) - x; then ask the
indefinite integral of 2*f(x)+x, and a double indefinite integral of 2*f(x) +
x, and a double indefinite integral of respect to both x and y. Change the
first line, and then run the program again.
IN: integrate(f(x_), x_) := f(x) - x
IN: integrate(2*f(x)+x, x)
OUT: 2*f(x) - 2*x +
1/2*x^2
IN: integrate(integrate(f(x)+y, x), y)
OUT: f(x)*y - x*y +
x*y^2
You can also ask SymbMath to perform
the following integrals:
integrate(integrate(f(x)+y^2,
x), y),
integrate(integrate(f(x)*y,
x), y),
integrate(x*f(x),
x),
triple integral of
f(x)-y+z, or others.
You continue to ask indefinite integral.
IN: integrate(integrate(f(x)+y, x from 0 to 1), y
from 0 to 2)
OUT: 2 f(1)
SymbMath can learn complicated derivatives
from a simple derivative, even though the function to be differentiated is an
unknown function, instead of standard function.
Example :
Differentiate f(x^2)^6, where f(x)
is an unknown function.
IN: d(f(x^2)^6, x)
OUT: 12 x f(x^2)^5
f'(x^2)
Output is only the part derivative.
f'(x^2) in the output suggest that you should teach SymbMath f'(x_). e.g. the
derivative of f(x) is another unknown function df(x), i.e. f'(x_) = df(x),
assign f'(x_) with df(x) and run it again.
IN: f'(x_) := df(x)
IN: d(f(x^2)^6, x)
OUT: 12 x f(x^2)^5
df(x^2)
This time you get the complete
derivative.
If you show SymbMath algebra, SymbMath can
learn integrals from that algebra.
Example :
Input f(x)^2=1/2-1/2*cos(2*x), then
ask for the integral of f(x)^2.
IN: f(x)^2 := 1/2-1/2*cos(2*x)
IN: integrate(f(x)^2, x)
OUT: 1/2 x - 1/4
sin(2 x)
SymbMath is very flexible. It
learned to solve these problems, even though the types of problems are
different, e.g. learning integrals from derivatives or algebra.
SymbMath has the ability to learn
complicated algebra from simple algebra.
Example:
Transform sin(x)/cos(x) into tan(x)
in an expression.
IN: sin(x)/cos(x) := tan(x)
IN: x+sin(x)/cos(x)+a
OUT: a + x + tan(x)
The difference between learning and
programming is as follows: the learning process of SymbMath is very similar to
the way human beings learn, and that is accomplished by knowing certain rule
that can be applied to several problems. Programming is different in the way
that the programmer have to accomplish many tasks before he can begin to solve
a problem. First, the programmer defines many subroutines for the individual
integrands (e.g. f(x), f(x)+y^2, 2*f(x)+x, x*f(x), etc.), and for individual
integrals (e.g. the indefinite integral, definite integral, the indefinite
double integrals, indefinite triple integrals, definite double integrals,
definite triple integrals, etc.), second, write many lines of program for the
individual subroutines, (i.e. to tell the computer how to calculate these
integrals), third, load these subroutines, finally, call these subroutines.
That is precisely what SymbMath do not ask you to do.
In one word, programming means that
programmers must provide step-by-step procedures telling the computer how to
solve each problems. By contrast, learning means that you need only supply the
necessary facts (usually one f’ and/or one integral of f(x)), SymbMath will
determine how to go about solutions of many problems.
If the learning is saved as a
library, then you do not need to teach SymbMath again when you run SymbMath
next time.
SymbMath is an interpreter, and runs a
SymbMath program in the Input window, which is written by any editor in the
text (ASCII) file format.
SymbMath language is a procedure
language, which is executed from top to bottom in a program, like BASIC,
FORTRAN, or PACSAL. It also is an expression-oriented language and functional
language.
The SymbMath program consists of a
number of statements. The most useful statement contains expressions, the
expression includes data, and the most important data is functions.
The structure of SymbMath language
is:
data -> expression
-> statement -> program
Note that upper and lower case
letters are different in SymbMath language, (e.g. abc is different from ABC)
until the switch lowercase := on.
In the following examples, a line of
"IN: " means input, which you type in the Input window, then leave
the Input window by <Esc>, finally run the program by the command
"Run"; while a line of "OUT:" means output. You will see both
input and output are displayed on two lines with beginning of "IN: "
and "OUT: " in the Output window. You should not type the word
"IN: ". Some outputs may be omit on the examples.
#
is a comment statement.
You can split a line of command into multi-lines
of command by the comma ,. The comma without any blank space must be the last
character in the line.
The data types in SymbMath language is the
numbers, constants, variables, functions, equations, arrays, array index,
lists, list index, and strings. All data can be operated. It is not necessary
to declare data to be which type, as SymbMath can recognise it.
The types of numbers are integer, rational,
real (floating-point), and complex numbers in the range from -infinity to
infinity.
In fact, the range of the input real
numbers is
-inf,
-(10^300)^(10^300) to -10^(-300), 0, 10^(-300) to (10^300)^(10^300), inf.
The range of the output real numbers
is the same as input when the switch numeric := off, but when the switch
numeric := on, it is
-inf, -1.E300 to -1.E-300, 0,
1.E-300 to 1.E300, inf.
It means that the number larger than
1.e300 is converted automatically to inf, the absolute values of the number
less than 1.e-300 is converted to 0, and the number less than -1e300 is
converted to -inf.
For examples:
-------------------------------------------
Numbers Type
23 integer
2/3 rational
0.23 real
2.3E2 real
2+3*i complex
2.3+i complex
---------------------------------------------
That "a" and "b"
are the same means a-b = 0, while that they are different means a-b <> 0.
For the real numbers, the upper and
lower case letters E and e in exponent are the same, e.g. 1e2 is the same as
1E2.
The constants are the unchangeable values.
There are some built-in constants. The name of these built-in constants should
be avoided in the user-defined constants.
------------------------------------------------------------------
Built-in Constants Meanings
pi:=3.1415926536 the circular constant.
e:=2.7182818285 the base of the natural
logarithms.
i:=sqrt(-1) the imaginary
sign of complex numbers.
inf infinity.
-inf negative
infinity.
c_inf complex
infinity, both real and imaginary parts
of
complex numbers are infinity. e.g. inf+inf*i.
constant the integral
constant.
discont discontinuity,
e.g. 1/0. (You can evaluate the one-sided value by
x=x0+zero or x0-zero if the value of expression is discont).
x0-zero to evaluate
left-sided value when x approach x0
from
negative (-inf) direction, as zero -> 0.
x0+zero to evaluate
right-sided value when x approach x0
from
positive (+inf) direction, as zero -> 0.
undefined the undefined
value, e.g. indeterminate forms:
0/0,
inf/inf, 0*inf, 0^0, etc.
--------------------------------------------------------------------
Zero is the positive-directed 0, as
the built-in constant. f(x0+zero) is the right-hand sided function value when x
approaches to x0 from the positive direction, i.e. x = x0+. f(x0-zero) is the
left-sided function value when x approaches to x0 from the negative direction,
i.e. x = x0-. e.g. f(1+zero) is the right-hand sided function value when x
approaches to 1 from the positive (+infinity) direction, i.e. x = 1+, f(1-zero)
is the left-hand sided function value when x approaches to 1 from the negative
(-infinity) direction, i.e. x = 1-; exp(1/(0+zero)) gives inf, exp(1/(0-zero))
gives 0.
The inf, discont and undefined can
be computed as if numbers.
Example:
IN: inf+2, discont+2, undefined+2
OUT: inf, discont,
undefined
Notice that the discont and
undefined constants are different. If the value of an expression at x=x0 is
discont, the expression only has the one-sided value at x=x0 and this one-sided
value is evaluated by x=x0+zero or x=x0-zero. If the value of an expression at
x=x0 is undefined, the expression may be evaluated by the function lim().
Example: evaluate exp(1/x) and
sin(x)/x at x=0.
IN: f(x_) := exp(1/x)
OUT: f(x_) :=
exp(1/x)
IN: f(0)
OUT: discont # f(0) is discontinuity, only has
one sided value
IN: f(0+zero) # right-sided value
OUT: inf
IN: f(0-zero) # left-sided value
OUT: 0
IN: subs(sin(x)/x, x = 0)
OUT: undefined
IN: lim(sin(x)/x, x = 0) #
it is evaluated by lim()
OUT: 1
The sequence of characters is used as the
name of variables. Variable names can be up to 128 characters long. They must
begin with a letter and use only letters and digits. SymbMath knows upper and lower case distinctions
in variable names, so AB, ab, Ab and aB are the different variables. They are
case sensitive until the switch lowercase is set to on (i.e. lowercase := on).
Variables can be used to store the
results of calculations. Once a variable is defined, it can be used in another
formula. Having defined X as above, you could define Y := ASIN(X). You can also
redefine a variable by storing a new value in it. If you do this, you will lose the original
value entirely.
Assign a result to a variable, just
put
<var-name> := expression
e.g. x := 2 + 3 # assign value to x
Variables can be used like constants
in expressions.
For example:
a := 2 + 3
b := a*4
If an undefined variable is used in
an expression, then the expression returns a symbolic result (which may be
stored in another variable). Pick an
undefined variable name, say x, and enter:
y := 3 + x # formula results since x undefined
x := 4 # Now x is defined
y # y returns 7, but its value is still the
formula 3 + x
x := 7 # revalue x
y # new value for y
Note that in symbolic computation,
the variable has not only a numeric value but also a symbolic value.
Symbolic values for variables are
useful mostly for viewing the definitions of functions and symbolic
differentiation and integration.
Watch out for infinite recursion
here. Defining
x := x+3
when x has no
initial value, it will not cause an immediate problem, but any future reference
to x will result in an infinite
recursion !
A value can be assigned to the
variable, by one of three methods:
(1) the assignment :=,
(2) the user-defined function f(),
(3) subs(y, x = x0).
e.g.
y:=x^2
x:=2 # assignment
y
f(2) # if f(x) has
been defined, e.g. f(x_):=x^2.
subs(x^2, x = 2) # evaluate x^2 when x =
2.
The variable named last is the
built-in as the variable last is always automatically assigned the value of the
last output result.
The usual used independent variable
is x.
By default, |x| < inf and all
variables are complex, except that variables in inequalities are real, as usual
only real numbers can be compared. e.g. x is complex in sin(x), but y is real
in y > 1.
You can restrict the domain of a
variable by assuming the variable is even, odd, integer, real number, positive
or negative (see Chapter Simplification and Assumption).
Patterns stand for classes of expressions.
_ any expression.
x_ any expression, given the name x.
Patterns should appear on the
left-hand side of the assignment only, not on the right-hand side of the
assignment. Patterns are only used in definition of functions, procedures and
rules.
Patterns are used to define
functions and rules for pattern match.
These are two types of functions: internal
and external. The internal function is compiled into the SymbMath system. The
external function is the library written in SymbMath language, which is
automatically loaded when it is needed. (See Chapter Library and Package). The
usage of both types are the same. You can change the property or name of the
external function by modifying its library file, or you add a new external
function by creating its library file, but you cannot change the internal
function.
Different versions of SymbMath have
different number of standard mathematical functions. The Advanced Version C has
all of them. See the following table in detail for other versions. All below
standard functions, (except for random(x), n!, fac(n) and atan2(x,y)), can be
differentiated and integrated symbolically.
Table 17.1.5.1
Standard Mathematical Functions
-----------------------------------------------------------------------
Functions Meanings
random(x) generate a random number.
n! factorial of n.
fac(n) the same as n!.
sqrt(x) square root, the same
as x^0.5.
root(x,n) all n'th root of x.
exp(x) the same as e^x.
sign(x) 1 when re(x) > 0, or
both re(x) = 0 and im(x) > 0; 0 whenx=0;
-1
otherwise.
abs(x) absolute value of x.
ln(x) natural logarithmic
function of x, based on e.
log10(x)
sin(x) sine function of x.
cos(x)
............................... above functions in Shareware Version A
...............
tan(x)
csc(x)
sec(x)
cot(x)
asin(x) arc sine function of x, the
inverse of sin(x).
acos(x)
atan(x)
acot(x)
asec(x)
acsc(x)
atan2(x,y)
.............................
above functions in Student Version B .................
sinh(x) hyperbolic sine function of x.
cosh(x)
tanh(x)
csch(x)
sech(x)
coth(x)
asinh(x) arc hyperbolic sine
function of x, the inverse of sinh(x).
acosh(x)
atanh(x)
acoth(x)
asech(x)
acsch(x)
--------------------------------------------------------------------------
Calculus functions are for calculus
calculation. The first argument of the function is for evaluation, and the
second argument is a variable that is with respect to.
Table 17.1.5.2 Calculus Functions
--------------------------------------------------------------------------------------------------------------
Functions Meanings
replace(y, x , x0) evaluates y when x = x0.
lim(y, x = x0) gives the limit of y when x approaches x0.
Note that the correct answers usually for the
indeterminate forms: 0/0, inf/inf, 0*inf, 0^0, inf^0.
d(y, x) differentiate y with respect to x.
d(y, x, order) gives the n-th order derivative of y with
respect to an undefined variable x.
d(y) implicit differentiation, the default variable is x .
integrate(y, x) find the indefinite integral of y with
respect to an undefined variable x.
integrate(y,x,a,b) find the definite integral of y with
respect to an undefined variable x taken from
x=a to x=b.
integrate(y,x,a,b,c) find the definite integral of y with
respect to an undefined variable x taken from
x=a to x=b, then to x=c, where b is singularity.
integrate(y, x , a ,
b) the
same as integrate(y,x,a,b).
integrate(y) implicit integration, the default variable is x.
dsolve(y'=f(x,y), y,
x) solve a differential equation.
sum(y,x) indefinite sum with a variable x.
sum(y) indefinite sum y, the default variable is x.
sum(y, x , xmin ,
xmax) partial sum of y from xmin to xmax .
sum(y,x,0, inf) infinite sum.
sum(y, x , xmin ,
xmax , dx) sum of y, by
step dx.
prod(y, x , xmin ,
xmax) product
of y step=1.
prod(y, x , xmin ,
xmax , dx) product of y.
----------------------------------------------------------------------------------------------------------------
If a second argument x is omitted in
the functions d(y) and integrate(y), they are implicit derivatives and
integrals. If f(x) is undefined, d(f(x), x) is differentiation of f(x). These
are useful in the differential and integral equations. (see later chapters).
For examples:
integrate(integrate(F,x), y) is
double integral of F with respect to both variables x and y.
d(d(y,x),t) is the mixed derivative
of y with respect to x and t.
Examples:
--------------------------------------------------------------------------------
differentiation d() d(x^2,x)
integration integrate() integrate(x^2,x)
limit lim() lim(sin(x)/x, x = 0)
---------------------------------------------------------------------------------
Table 17.1.5.3.1 The is*(x) Functions
---------------------------------------------------------------------
Function Meaning
isodd(x) test if x is
an odd number.
iseven(x) test if x is an even
number.
isinteger(x) test if x is an integer
number.
isratio(x) test if x is a rational
number.
isreal(x) test if x is
a real number.
iscomplex(x) test if x is a complex number.
isnumber(x) test if x is a number.
islist(x) test if x is a list.
isfree(y,x) test if y is free of x.
issame(a,b) test if a is the same as b.
islarger(a,b) test if a is larger than b.
isless(a,b) test if a is less than b.
----------------------------------------------------------------------
All of the is* functions give either
1 if it is true or 0 otherwise.
The type(x) gives the type of x. Its
value is a string.
Table 17.1.5.3.2 The type(x) functions
--------------------------------------------------
x type(x)
1 integer
1.1 real
2/3 ratio
1+i complex
sin(x) sin()
[1,2] []
a symbol
"a" string
a+b +
a*b *
a^b ^
a=b =
a==b ==
a>b >
a>=b >=
a<b <
a<=b <=
a<>b <>
a,b ,
---------------------------------
You also can test x, e.g. if x is
type of real number, by
type(x)=="real"
Table 17.1.5.4.1 Algebra Functions
---------------------------------------------------------------------------------------------------
expand(F) expand (a+b)^2 to a^2 +
2*a*b + b^2.
factor(F) factorise a^2 + 2*a*b +
b^2 to (a+b)^2.
solve(f(x)=0, x) solve polynomial and systems of
linear equations
---------------------------------------------------------------------------------------------------
Note: the Shareware
Version has not solve().
For example:
-----------------------------------------------------------------------
solving solve() solve(x^2+1 = 0, x)
expanding expand() expand((a+b)^2)
factoring factor() factor(a*c+b*c)
----------------------------------------------------------------------
Conversion functions convert a type
of data to another type of data.
Table 17.1.5.4.2 Conversion Functions
---------------------------------------------------------------------
listsum([a,b]) convert list to sum.
coef(expr, x^2) gives the coefficient of x^2 in
expr.
left(x^2=b) left hand side of an
equation.
right(x^2=b) right hand side of an
equation.
re(x) real part of
complex numbers.
im(x) imaginative part of
complex numbers.
num(x) convert x to the
floating-point number.
ratio(x) convert x to the
rational number.
round(x) convert x to the rounded
integer.
trunc(x) convert x to
the truncated integer.
----------------------------------------------------------------------
Table 17.1.5.4.3 The List and Table Functions
-------------------------------------------------------------------------------------------
list(f(x), x from
xmin to xmax step dx) lists of f(x).
table(f(x), x from
xmin to xmax step dx) data table of function
values.
--------------------------------------------------------------------------------------------
Above functions can be operated and
chained, like the standard functions.
You can define the new functions, which
include the standard functions, calculus functions, and algebraic operators.
Define a new function f(x) by
f(x_) := x^2
and then call f(x)
as the standard functions. The function name can be any name, except for some
keywords. (for the maximum number of arguments, see Chapter System Limits).
Clears a variable or function from
assignment by
clear(x) # clear x from assignment.
clear(f(x)) # clear f(x) from assignment.
clear(a>0) # clear a>0 from assignment.
Variables can be used in function
definitions. It leads to an important difference between functions and
variables. When a variable is defined,
all terms of the definition are evaluated.
When a function is defined, its terms are not evaluated; they are
evaluated when the function is evaluated. That means that if a component of the
function definition is changed, that change will be reflected the next time the
function is evaluated.
A procedure is similar to a function, but
the right side of assignment in its definition is multi statements grouped by
block(). The block(a,b,c) groups a,b,c and only returns the last argument as
its value, or returns the second last argument as its value if the last
argument is local(). It is used as grouper in definition of a procedure. All
variables in block are local.
e.g. f(x_):=block(p:=x^6,p)
Remember that you can split a line
of program into multi-lines program at comma,.
A rule is similar to a function. In
definition of function, all arguments of function are simple variables, but in
definition of rules, the first argument may be a complicated expression.
e.g.
f(x_,y_) := x^2+y^2 # defining
function
f(x_,y_) := block(a:=2, a+x^2+y) # defining procedure
log(x_ * y_) := log(x)+ log(y) # defining rule
An equation is an equality of two sides
linked by an equation sign =, e.g. x^2+p = 0, where the symbol = stands for an
equation. Note that the symbols "=", "==" and
":=" are different: ":=" is the assignment, "=="
is the equal sign, but "=" is the equation sign.
Example:
IN: 2 = 2
OUT: 2 = 2 # unevaluated
IN: 2 == 2
OUT: 1 # evaluated to 1 (true)
Systems of equations are a list of
equations, e.g.
[a1*x+a2*y=a3,
b1*x+b2*y=b3].
e.g.
a < b less than
a <= b less than or equal to
a > b greater than
a >= b greater than or equal to
a == b equal to
a <> b not equals
Lists are similar to lists in such language
as PROLOG.
[a, b, c] is a list.
[a, b, [c1, c2]] is a list of lists.
The list index is the index for n-th
element in a list. e.g. b[2] indicates the second element in the list b.
The built-in list index is
last[number]. The name of last output list is last, e.g. last[1] is the first
element in the last output list.
Arrays are the same as those in such
language as PASCAL and FORTRAN. But you can use arrays without declaring
arrays, unlike in PASCAL and FORTRAN.
e.g.
a[1]:=1
a[2]:=4
The array index is the index for
n-th element in an array. e.g. a[2] indicates the second element in the array
a.
A string is a sequence of characters
between two quotation marks. e.g. "1234567890". Note that 1234 is
number but "1234" is string. The number can be calculated and only
has 11 of max digits, while string cannot be calculated and has 64000 of max
characters long.
Note that the output of strings in
SymbMath is without two quotation marks. This makes text output to graph and
database more readable.
Strings can be stored in variables,
concatenated, broken, lengthen, and converted to numbers if possible.
e.g.
IN: p := "abc" # "abc" is stored in variable p
OUT: p := abc
IN: concat("s","t") # concatenate "s" and
"t"
OUT: st
IN: length("abc") # count length of "abc"
OUT: 3
IN: number("123") # convert string "123" into
number 123
OUT: 123
IN: type(a), type("a")
OUT: symbol, string
The expressions (i.e. expr) are made up of
operators and operands. Most operator are binary, that is, they take two
operands; the rest are unitary and take only one operand. Binary operators use
the usual algebraic form, e.g. a+b.
There are two kinds of expressions:
numeric and Boolean. The numeric expression is combination of data and algebraic
operators while the Boolean expression is combination of data and relational
operators and logic operators. These two kinds of expressions can be mixed, but
the numeric expression has higher priority than Boolean operators. x*(x>0)
is different from x*x>0. x*x>0 is the same as (x*x)>0.
e.g.
a+b+3 numeric expression,
a>0 Boolean expression
a>0 and b>0 Boolean expression
(x>0)*x mixed numeric and Boolean
expression
Table 17.2.1 Operators
--------------------------------------------------------------------------------------------------
Operation Operators Examples Order
comma , a:=2, b:=3 1
assignment := p:=2+3 2
and and a>2 and a<8 2
or or a>2 or b>2 2
equation = x^2+x+1 = 0 3
equal == a==2 3
larger than > a>2 3
larger and equal >= a>=2 3
less than < a<2 3
less and equal <= a<=2 3
unequal <> a<>2 3
plus + a+b 4
minus - a-b 4
mutilation * a*b 5
division / a/b 5
power ^ a^b 6
factorial ! n! 6
positive + +a 7
negative - -a 7
function f() sin(x) 7
list index f[] f[1] 7
parentheses () (a+b)*c 7
list [] [a,b] 7
------------------------------------------------------------------------------------------------
All functions have the same 7th
order.
Operations with higher order
precede, otherwise operations with equal precedence are performed from left to
right. These are the usual algebraic
conventions.
a^b^c is the same as (a^b)^c.
You can get operators by type(x).
-------------------------------------------------------------------------------------------
plus + a+b 4
minus - a-b 4
mutilation * a*b 5
division / a/b 5
power ^ a^b 6
fractorial ! a! 6
---------------------------------------------------------------------------------------------
Before you can write loops, you must be
able to write statements that evaluate to 1 or 0, and before you can do that,
you must be able to write useful statements with logical values. In
mathematics, these are relational statements.
SymbMath allows you to compare
numbers six ways:
a < b less than
a <= b less than or equal to
a > b greater than
a >= b greater than or equal to
a == b equal to
a <> b not equals
SymbMath uses the double equals sign
== (like C language) for "is equal to" to distinguish this operator
from the equation =.
The result of a comparison of two
real numbers is either 1 or 0. If the comparison is not both real numbers, it
left unevaluated.
SymbMath uses the logical operators: AND, and OR.
You can combine comparison operators with them to any level of
complexity. In contrast to Pascal, logical operators in SymbMath have a lower
order or precedence than the comparisons, so a < b and c
> d works as expected. The result of
combining logical values with
logical operators is
another logical value (1 or 0). Bit
operations on integers can be performed using the same operations, but result
is integers.
SymbMath uses the
"short-circuit" definition of AND and OR when the arguments are
Boolean. Here are tables that show how
AND and OR are defined:
a and b
--------------------------------------------------------
b 1 0
a
1 1 0
0 0 0
------------------------------------------------------
a or b
--------------------------------------------------------
b 1 0
a
1 1 1
0 1 0
------------------------------------------------------
Short-circuit evaluation is used
because often one condition must be tested before another is meaningful.
The result of Boolean expression
with logic operators is either 1 or 0. Boolean expression like (1 < 3 or 1
> 4) return a real value 1 or 0.
Numeric expressions can replace Boolean ones, provided they evaluate to 1
or 0. The advantage here is that you can
define the step function that is 0 for x < a and 1 for x > a by entering:
step(x_, a_) := x > a
To define the function:
f(x) = x-1 if x < 1
= x^2-x if
x >= 1
enter:
f(x_) := (x-1)*(x < 1) + (x^2-x)*(x
>= 1)
These functions can be
differentiated and integrated symbolically.
A function call activates the function
specified by the function name. The function call must have a list of actual
parameters if the corresponding function declaration contains a list of formal
parameters. Each parameter takes the place of the corresponding formal
parameter. If the function is external, the function call will automatically
load the library specified by its function name plus extension when needed.
Some examples of the function calls
follow:
sin(x) # load the library sin when needed
integrate(x^2, x) # load the library int when needed
# is the comment statement sign.
You can add comments into a line, or
even produce a single line which is entirely a comment, by preceding the
comment sign with #.
For example:
# This is my program;
3 + 4 # My first calculation;
Comments make your calculations more
understandable, whether you are making a printed record of what you are doing
or if you just want to jot some notes to yourself while you are working.
The evaluation statement has the format:
expression
SymbMath evaluates any expression
which in a line and gives the value of the expression. e.g.
IN: 3 + 4
OUT: 7 # the value of 3+4
IN: d(x^6, x)
OUT: 6 x^5 # the value of d(x^6, x)
IN: subs(last, x = 1) # evaluate the last output when x = 1.
OUT: 6
The last output can be saved to a
variable for the later use by the built-in variable "last", e.g. f
:=last.
The assignment in SymbMath language is
similar to assignment in such language as PASCAL.
An assignment operator is :=
The assignment statement specifies
that a new value of expr2 be assigned to expr1, and saved into memory. The form
of the assignment statements is
expr1 := expr2;
You can use assignment for storing
result.
You can assign the result of
calculation or any formula to a variable with a command like: X := SIN(4.2).
The assignments are useful for long
calculations. You can save yourself a
lot of recalculations by always storing the results of your calculations in
your own variables instead of leaving them in the default variable last.
You can destroy the assignment to X with the
command clear(X). If X stored a large list, you could regain a considerable
amount of memory by clearing X. Also, since a variable and a function can have
the same name, you can clear a variable p, not a function p(x).
The assignment operator is also used
in the definition of a function or procedure.
Variables can be used in function
definitions, and that leads to an important difference between functions and
variables. When a variable is defined,
all terms of the definition are evaluated.
When a function is defined, its terms are not evaluated; they are
evaluated when the function is evaluated.
That means that if a component of the function definition is changed,
that change will be reflected the next time the function is evaluated.
e.g.
IN: p:=2+3 #
2+3 is evaluated at the time of assignment, p is assigned with 5.
OUT: p := 5
IN: p(x):=2+3 #
2+3 is evaluated when the value of p(x) is requested,
# p(x) is
assigned with 2+3.
OUT: p(x) := 2+3
If the left hand side of the
assignment is a variable, it is the immediate assignment (i.e. expr2 is
evaluated at the time of assignment); if the left hand side is a function, it
is the delayed assignment (i.e. expr2 is evaluated when the value of expr1 is
requested).
You can force all the components of
a function to be evaluated when the function is defined by preceding the
function with the command eval():
f(x_) := eval(2+3) # f(x_) is assigned with 5
Note that not only a variable but also
any expression can be assigned. e.g. x := 2, sin(x)/cos(x) := tan(x), a>0 :=
1.
Clear a variable, function or
expression from assignment by
clear(x) # clear x from assignment.
clear(f(x)) # clear f(x) from assignment.
clear(a>0) # clear a>0 from assume(a>0).
There are two conditional functions:
If(test , x)
If(test , x , y)
If(condition , x) gives x if condition evaluates to 1, or no
output otherwise.
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. The 2 words (then and else) can be replaced by
comma ,.
It is useful in definition of the
use-defined function to left the function unevaluated 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).
You can use two kinds of loops in SymbMath,
fixed length loops controlled by do() and variable-length loops controlled by
repeat(). The do() loop is similar to the FOR loop in BASIC language.
The control variable in the do()
loops is not limited to integer values. You can say:
do(f:=f+1, x from xmin
to xmax step dx)
It is similar to
FOR x := xmin TO xmax
STEP dx
f:=f+1
NEXT x
where xmin, xmax,
and dx are real values. If STEP dx is
omitted, it defaults to 1.
e.g.
IN: x:=0
OUT: x := 0
IN: do(x:=x+1, j from 1 to 5 step 1)
OUT: x := 5
The conditional loops are probably more useful than the do() loops if a
number of iteration is unknown. It is
repeat(y until test)
repeat(y, test)
The word (until) can be replaced by
comma ,. The repeat() repeats to evaluate f until the test is true (i.e. the
result of the test is 1).
Example:
IN: x:=1
OUT: x := 1
IN: repeat(x:=x+1 until x>5)
OUT: x := 6
The switch sets or changes the switch
status. The switch status is unchanged in memory until the switch is assigned
by the new value.
---------------------------------------------------------------------------------------------------------
Switch Action
output := math output form is math symbol
notation, this is default.
output := basic output form is BASIC format.
output := fortran output form is FORTRAN
format.
output := prolog output form is PROLOG format (internal
form).
output := off not display output.
output := on the same as output := basic
lowercase := on convert letters into the
lower-case letters.
lowercase := off not convert letters into
the lower-case letters, this is default.
numeric := on convert numbers to floating-point
numbers.
numeric := off not convert numbers to
floating-point numbers, this is default.
expand := on expansion. e.g. c*(a+b) to
c*a+c*b.
expand := off disable expansion, this is
default.
expandexp := on expand exponent. e.g. c^(a+b) to
c^a*c^b.
expandexp := off disable exponent expansion, this is
default.
------------------------------------------------------------------------------------------------------------------
When the switch output := math, the output
displays the math symbol notation.
When the switch output := basic, the
output form is the BASIC language format.
When the switch output := fortran,
the output form is the FORTRAN language format.
The default switch is output := math.
The output switch only changes the
output format, neither affects the internal format nor the calculation.
When the switch lowercase := on, all
letters are converted into the lower-case letters so the upper- and lower- case
letters are the same, e.g. EXP(x) is the same as exp(x).
Example:
IN: 1/2
OUT: ½
IN: numeric := on
IN: 1/2
OUT: 0.5
Example:
IN: a*(b+c)
OUT: a (b + c)
IN: expand := on
IN: a*(b+c)
OUT: a* b + a* c
Example:
IN: exp(a+b)
OUT: e^(a + b)
IN: expandexp := on
IN: exp(a+b)
OUT: e^a* e^b
The format of the readfile statement is
readfile("filename")
The filename is any MS-DOS file
name. If the file is not in the current directory, the filename should include
the directory. e.g.
readfile("directory\filename")
e.g. read a file named "int.sm":
readfile("int.sm")
It seems to copy the file into the
user program.
After a file is read, you can call
any part of this package from a second package, as it seems the part of the
first program has already been in the second program. you can read many files
into the SymbMath program at a time. However, all names of the variables are public
and name conflicts must be avoided.
Write a file to disk by
openfile("file")
do something
closefile("file")
Table 17.3.7. Reading and Writing Statements
---------------------------------------------------------------------
readchar read a character
from keyboard.
readline read a line
of strings from keyboard.
readfile("file") read (run or include) the file
named "file".
......................................................................
openfile("file") open the disk file
"file" for writing.
closedfile("file") closed the file "file".
writes(s) write s on screen, only
on graphics mode.
newline write next text on a
new line.
null not write.
block(a,b) write the value of the last
argument, b.
---------------------------------------------------------------------
where the filename
is any MS-DOS filename.
Note that the file must be closed by
the closefile() command when writing a file with the openfile() command, but
the file is automatically closed after reading the file. There must be the end
statement at the end of file for reading.
SymbMath can read expressions from a
disk file, then manipulate the expression, and finally write the result into
another disk file.
Example: an expression y:=x^6 is
already in the file "y.in",
The contents of the file
"y.in":
---------------------------------------------------------------------
y:=x^6
-----------------------------------------------------------------------
Run this SymbMath program
----------------------------------------------------------------------------
readfile("y.in") # read the expression y:=x^6 from
the file "y.in"
openfile("y.out") # open a disk file
"y.out" for writing
d(y,x) # differentiate y and
write the result to the file
closefile("y.out") # close the file and return output to
SymbMath
----------------------------------------------------------------------------
The contents of the file
"y.out":
---------------------------------------------------------------------
6*x^5
---------------------------------------------------------------------
In this way you can interface to
other software (see 3.7. Interface with Other Software).
These outputs in the disk file can
be edited in the Edit window or the Aux Edit window.
It is recommended to use the BASIC
output format by setting the switch output := basic when you write the output
into the disk file. The default switch is output := math.
You can executes a DOS command in SymbMath
language by
system("dir")
The all above statements are simple
statements. The sequence statement specifies that its component statements are
to be executed in the same sequence as they are written. They are separated by
the separators (comma ","). e.g.
a+b, 2+3
A library is a file of an external
function, which filename is its function name within 8 letters plus extension .
e.g. the library named sin is a file of the sin(x) function definition.
The library (the * file) is similar
to the MS-DOS *.BAT file. You do not need to load or read the library by any
command. SymbMath automatically load the library when it is needed. For
example, when you use the sin(x) function first time, the library sin will be
auto-loaded. The library must be in the default directory, otherwise the
library is not loaded and the function is not working. Only the assignments in
the library can be loaded, and others in the library will be omitted, but all
of these assignments will be not evaluated when they are loaded. You can clear
the library sin from memory by clear(sin(x)).
You can have libraries (external
functions) as many as your disk space available. You should use the "one
function per file" convenience.
Note that all names of the variables
in libraries and packages are public (global) except for those declared by
local() and name conflicts must be avoided.
A package is the SymbMath program
file which filename has not extension .
It is recommended that its filename has the extension .SM.
A package is similar to a library,
but the package must be read by a command
readfile("filename")
The filename can be any MS-DOS
filename. It is recommended that the filename is same function name used in
your program, plus the extension .sm. e.g. int.sm is the filename of the
integral package as the name of integral function is integrate(). If the file
is not in the current directory, the filename should include the directory.
e.g.
readfile("directory\filename")
After reading the package, you can
call the commands in the package from your program.
The readfile() command must be in a
single line anywhere.
Many packages can be read at a time.
You can convert a package of f.sm
into a library by renaming f.sm to f for auto loading, or a library f to a
package by renaming f to f.sm for not auto loading.
You can get help for all libraries
by the library Index command in the Help menu. You first open the library index
window by this command, then open a library by selecting its library name in
the library index window.
There are many libraries and
packages. The following are some of them.
Table 17.4.1 Libraries
------------------------------------------------------------------------
File Name Function
plot plotting
functions.
d derivatives.
int integrals.
sin sin(x) function.
cos cos(x) function.
fac n!.
sign sign(x) function.
abs abs(x) function.
arg arg(x) function.
sum sum function.
NInte numeric integration.
NSolve numeric solver of equation.
DSolve differential equation solver.
gamma gamma function.
ei exponential
integral function.
series Taylor series.
partSum partial sum.
infSum infinite sum.
sinh sinh(x) function.
cosh cosh(x) function.
average average([x]) function.
listSum sum of list of data.
react chemical reactions
database.
phoneNo phone No. database.
------------------------------------------------------------------------
Table 17.4.2 Packages
------------------------------------------------------------------------
init.sm initial package when
running the program in
the Input
window. It contains switches on the default status.
chemical.sm the atomic weight of chemical
elements.
ExpandTr.sm expansion of trig functions.
expandLn.sm expand ln(x*y).
ExpandGa.sm expand gamma(n,x).
ExpandEi.sm expand ei(n,x).
units.sm an units conversion.
listPlot.sm plotting a list of data.
plotData.sm interfacing software PlotData.
-----------------------------------------------------------------------
When a program is run in the Input window,
SymbMath first automatically reads (or runs) the initial package
"init.sm". The commands in the "init.sm" package seems to
be the SymbMath system commands. You can read other packages (e.g. f.sm) in the
initial package "init.sm", so the commands in the package
"f.sm" seems to be in SymbMath system. You do this by adding the
readfile("f.sm") into the init.sm file:
readfile("f.sm")
The lnexpand.sm package does the
logarithmic expansion. e.g. ln(a*b) is expanded into ln(a)+ ln(b).
SymbMath recognises 100 symbols of chemical
elements and converts them into their atomic weights after the chemical package
of "Chemical.sm" is read.
Example:
Calculate the weight percentage of
the element C in the molecule CH4.
IN: readfile("chemical.sm")
IN: numeric := on
IN: C/(C+H*4)*100*%
OUT: 74.868 %
Example:
Calculate the molar concentration of
CuO when 3 gram of CuO is in 0.5 litre of a solution.
IN: readfile("chemical.sm")
IN: numeric := on
IN: g:=1/(Cu+O)*mol
IN: 3*g/(0.5*l)
OUT: 0.07543 mol/l
You can run SymbMath from another software
as a engine. Anthoer software sends a text file to SymbMath, then run SymbMath
in background, get result back from SymbMath.
Interface
with other software, (e.g. CurFit, Lotus 123) is similar to interface with the
software PlotData in the plotdata package "plotdata.sm".
After load the file
"plotdata.sm", the functions
plotdata(y, x)
plotdata(y, x from xmin
to xmax)
plotdata(y, x from xmin
to xmax step dx)
plot a function of y
by mean of the software PlotData. The plotdata() first opens a file
"SymbMath.Out" for writing, then write the data table of the y
function into the file "SymbMath.Out", then close the file, and
finally call the software PlotData to plot. These are done automatically by
plotdata(). After it exits from PlotData, it automatically return to SymbMath.
When SymbMath is interfaced with the
software PlotData, SymbMath produces the data table of functions, and PlotData
plots from the table. So SymbMath seems to plot the function. This interface
can be used to solve equations graphically.
Example:
plot x^2 by
interfacing software PlotData.
IN: readfile("plotdata.sm")
IN: plotdata(x^2, x)
in the software
PlotData, you just select the option to read the file "SymbMath.Out"
and to plot. PlotData reads the data in the SymbMath format without any
modification (and in many data format).
In PlotData,
in the main menu:
1 <Enter>
in the read menu:
2 <Enter>
<Enter>
in the main menu:
2 <Enter>
in the graph menu:
1 <Enter>
where <Enter>
is the <Enter> key. Refer to PlotData for detail.
SymbMath
includes extensive facilities for graphing. It supports BGI graphics, which
graphics commands are the same as those in Borland Turbo Pascal and Turbo C,
except for:
different
graphics commands
---------------------------------------------------------------
SymbMath Turbo
Pascal or C
graph initgraph(drive,mode,path)
text closegraph
writes(x) outtext(x)
-----------------------------------------------------------------
Note that if your monitor is Hercules, you
must load the MSHERC.COM program as a TRS program before you run PlotData.
Otherwise you will get Error when you plot.
Before
graphing (drawing or plotting), you must initialize the graphics system and
puts the computer hardware into graphics mode by the command:
graph
then you can draw a line by
line(x1,y1,x2,y2)
lineto(x2,y2)
linerel(dx,dy)
draw a circular arc by
arc(x,y,angle1,angle2,radius)
draw an elliptical arc by
ellipse(x,y,angle1,angle2,XRadius,YRadius)
put a pixel by
putpixel(x,y,color)
you
can move a pointer by
moveto(x,y)
moverel(dx,dy)
A
upper left corner on your graphics screen coordinates is (0,0).
The
style of the line can be set by
setlinestyle()
You
can set screen colors. If you are using a color system, you can set the color
of the next line or graph or text with the command:
setcolor(color)
where color is color number (i.e. an integer in
the range 0..15), or one of the
colorno(x) function value.
You
can set the background color for your graphs with the command
setbkcolor(color)
where color is color number (i.e. an integer in
the range 0..15), or one of the
colorno(x) function value.
Table 18.1 ColorNo(x) function
in the colorno library
--------------------------------------------------------------------------------------
x value
black 0
blue 1
green 2
cyan 3
red 4
magenta 5
brown 6
lightgray 7
gray 8
lightblue 9
lightgreen 10
lightcyan 11
lightred 12
lightmagenta 13
yellow 14
white 15
-----------------------------------------------------------------------
You
can set line styles.
On
both monochrome and color systems, you can draw lines and graphs with different
line styles. (Since the line segments
used to draw graphs are usually very short, different line styles may not be
distinguished in graphs, but they will be distinguished on long lines.) Linestyles are indicated by integers in the
range 0..3, and are set by the command:
setlinestyle(style,u,thickness)
where style, u and thickness are integers.
You
can set the text style by
settextstyle(font,direction,size)
where font, direction and size are integers.
You
can add labels to your graphs by
writes(s)
You
can put alphanumeric labels anywhere on your graphic screens. They can be horizontal
or vertical, and they can be printed in various sizes. To print a string s
horizontally on the screen with the lower-left corner at the screen
coordinates (x,y), use two commands:
moveto(x,y),
writes(s)
To write vertically bottom to top, use two
commands:
settextstyle(1,2,2),
writes(s)
If
SymbMath attempts to graph a point (x,y) which is outside the the screen
coordinate, it ignores the point and continues.
No error message is generated, and even functions which are undefined on
part of the graphing domain can be graphed.
You
can get the max x and max y on your graphics screen
coordinates by
getmaxx
getmaxy
You
can get the current point(x, y) on your graphics screen coordinates by
getx
gety
You
can get the background color and foregroud color on your graphics screen by
getbkcolor
getcolor
You
can read a character from the keyboard or pause by the command:
readchar
You
can clear graph by
cleardevice
SymbMath
auto goes back the text mode at the end of run. You can force it goes back the
text mode by the command:
text
Example:
# drawing
a group of circles and ovals.
# Circles
are 9 planets around sun.
graph #
graph mode
do(circle(getmaxx*0.5+2.5*x,getmaxy*0.5,5),
x,0,90,10)
do(oval(getmaxx*0.5,getmaxy*0.5,2.5*x,x),
x,10,90,10)
readchar #
pause graph by read a char
text #
back text mode
You can plot a function of y = f(x) on the
xy-plane by external function:
plot(f(x),x)
plot(f(x),x,xmin,xmax)
plot(f(x),x,xmin,xmax,ymin,ymax)
plot(f(x),x,xmin,xmax,ymin,ymax,color)
f(x) can be either a
function with bound variable x or an
expression involving x. For example, you
could graph the parabola with the command plot(x^2,x).
The xmin and xmax are range of
x-axis, the ymin and ymax are range of y-axis. The default values are xmin=-5,
xmax=5, ymin=-5, and ymax=5. The values of xmin, xmax, ymin, ymax are real
numbers, such that xmin < xmax and
ymin < ymax. Thses values tell
SymbMath that the visible screen corresponds to a portion of the xy-plane
with xmin <= x <= xmax and
ymin <= y <= ymax.
The operator plot() plots one point
(x,f(x)) for each pixel on the x-axis, and connects successive points. To omit the connections and just plot the
points, use the command:
dotplot(f(x),x)
To plot only every 20th point, which
is useful for rapidly graphing complicated functions, use
sketch(f(x),x)
If you want your circles and squares
to look correct --that is, if you want one vertical unit to be really the same
distance as one horizontal unit--you should select window parameters so that
the horizontal axis is 1.4 times as long as the vertical axis.
Example 5.2:
plot x^3 and sin(x).
IN: graph
IN: plot(x^3,x)
IN: plot(sin(x),x)
You can plot the parametric functions of
x=x(t) and y=y(t) by
paraplot(x(t),y(t),t)
paraplot(x(t),y(t),t,tmin,tmax)
paraplot(x(t),y(t),t,tmin,tmax,ymin,ymax)
Example:
IN: graph
IN: paraplot(sin(t),sin(2*t),t)
You can graph the portion of a polar
curve r = f(t) that lies in the window
with external function:
polaplot(f(t),t)
polaplot(r, t,tmin,tmax)
polaplot(r,
t,tmin,tmax,rmin,rmax)
f(t) can be a
function with bound variable t or an expression involving t.
Example:
Plot a circle r = 1 in polar coordinates.
IN: graph
IN: polaplot(1,t)
The variable t
covers the domain (0, 2*pi); you can change this default by specifying a range
for t:
polaplot(1, t,0,pi)
You can plot data by
dataplot([x1,x2,...],
[y1,y2,...])
dataplot([x1,x2,...],
[y1,y2,...], xmin,max)
dataplot([x1,x2,...],
[y1,y2,...], xmin,xmax,ymin,ymax)
dataplot([x1,x2,...],
[y1,y2,...], xmin,xmax,ymin,ymax,link)
Example:
IN: graph
IN: dataplot([1,2,3], [1,4,9])
You can plot a list of data by
listplot([y1,y2,...])
for x1=1, x2=2,
x3=3, ....
Example:
IN: graph
IN: listplot([1,4,9])
You can fit a set of data into a
function x^2
Example 5.7.
IN: graph
IN: dataplot([1,2,3,4],[1,4,9,16],0,5,0,20)
IN: plot(x^2,x,0,5,0,20)
1. Load the
Graphics.EXE file into memory in the DOS prompt by
c:\DOS\graphics
2. Run SymbMath,
then display a graph on screen.
3. Print Screen by
pressing the <Print Screen> key on keyboard.
Note that you must
load the Graphics.EXE file before you run SymbMath.
SymbMath is a multi-windowed editor in which you can
copy-and-paste anywhere in a file and between files, even from the Help window.
When you start SymbMath, the
computer screen looks like:
-----------------------------------------------------------------------------------------
| File Input
Run Output Color Help Example Keyword |
----------------------------------
Input -----------------------------------------------
|
|
|
|
| |