Intro to MATLAB symbolic features.
% Introduction to the Symbolic Math Toolbox.
% The Symbolic Math Toolbox uses "symbolic objects" produced
% by the "sym" funtion. For example, the statement
x = sym('x');
% produces a symbolic variable named x.
% You can combine the statements
a = sym('a'); t = sym('t'); x = sym('x'); y = sym('y');
% into one statement involving the "syms" function.
syms a t x y
% You can use symbolic variables in expressions and as arguments to
% many different functions.
r = x^2 + y^2
theta = atan(y/x)
e = exp(i*pi*t)
exp(i*pi*t) % Notice that it did not evaluate numerically this expression since
% one term in it, the t, is symbolic. Even though i and pi are numeric.
% It is sometimes desirable to use the "simple" or "simplify" function
% to transform expressions into more convenient forms.
f = cos(x)^2 + sin(x)^2
f = simple(f)
% Derivatives and integrals are computed by the "diff" and "int" functions.
% Use pretty command to format the output of symbolic computation to make it easy
% to read
1/2 pi erf(t)
% If an expression involves more than one variable, differentiation and
% integration use the variable which is closest to 'x' alphabetically,
% unless some other variable is specified as a second argument.
% In the following vector, the first two elements involve integration
% with respect to 'x', while the second two are with respect to 'a'.
[int(x^a), int(a^x), int(x^a,a), int(a^x,a)]
[ x^(a+1)/(a+1), 1/log(a)*a^x, 1/log(x)*x^a, a^(x+1)/(x+1)]
% You can also create symbolic constants with the sym function. The
% argument can be a string representing a numerical value. Statements
% like pi = sym('pi') and delta = sym('1/10') create symbolic numbers
% which avoid the floating point approximations inherent in the values
% of pi and 1/10. The pi created in this way temporarily replaces the
% built-in numeric function with the same name.
pi = sym('pi')
delta = sym('1/10')
s = sym('sqrt(2)')
% Conversion of MATLAB floating point values to symbolic constants involves
% some consideration of roundoff error. For example, with either of the
% following MATLAB statements, the value assigned to t is not exactly one-tenth.
t = 1/10, t = 0.1
% The technique for converting floating point numbers is specified by an
% optional second argument to the sym function. The possible values of the
% argument are 'f', 'r', 'e' or 'd'. The default is 'r'.
% 'f' stands for 'floating point'. All values are represented in the
% form '1.F'*2^(e) or '-1.F'*2^(e) where F is a string of 13 hexadecimal
% digits and e is an integer. This captures the floating point values
% exactly, but may not be convenient for subsequent manipulation.
%'r' stands for 'rational'. Floating point numbers obtained by evaluating
% expressions of the form p/q, p*pi/q, sqrt(p), 2^q and 10^q for modest sized
% integers p and q are converted to the corresponding symbolic form. This
% effectively compensates for the roundoff error involved in the original evaluation,
% but may not represent the floating point value precisely.
% If no simple rational approximation can be found, an expression of the form
% p*2^q with large integers p and q reproduces the floating point value exactly.
% 'e' stands for 'estimate error'. The 'r' form is supplemented by a term
% involving the variable 'eps' which estimates the difference between the
% thoretical rational expression and its actual floating point value.
% 'd' stands for 'decimal'. The number of digits is taken from the current
% setting of DIGITS used by VPA. Fewer than 16 digits looses some accuracy,
% while more than 16 digits may not be warranted.
% The 25 digit result does not end in a string of 0's, but is an accurate
% decimal representation of the floating point number nearest to 1/10.
% MATLAB's vector and matrix notation extends to symbolic variables.
n = 4;
A = x.^((0:n)'*(0:n))
[ 1, 1, 1, 1, 1]
[ 1, x, x^2, x^3, x^4]
[ 1, x^2, x^4, x^6, x^8]
[ 1, x^3, x^6, x^9, x^12]
[ 1, x^4, x^8, x^12, x^16]
D = diff(log(A))
[ 0, 0, 0, 0, 0]
[ 0, 1/x, 2/x, 3/x, 4/x]
[ 0, 2/x, 4/x, 6/x, 8/x]
[ 0, 3/x, 6/x, 9/x, 12/x]
[ 0, 4/x, 8/x, 12/x, 16/x]
% Example to display irrational numbers to an arbitrary precision
Plotting functions for symbolic expressions:
Ezplot, ezcontour, ezcontourf, ezmesh, ezmeshc, ezplot3, ezpolar, ezsurf, ezsurfc
>> ezplot(' x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1',[0.985,1.01])
MATLAB commands related to symbolic manipulation
SUBEXPR Rewrite in terms of common subexpressions.