Applied Numerical Methods Using MATLAB, 2nd Edition

Appendix GSymbolic Computation

Chapter Outline

• G.1 How to Declare Symbolic Variables and Handle Symbolic Expressions
• G.2 Calculus
  • G.2.1 Symbolic Summation
  • G.2.2 Limits
  • G.2.3 Differentiation
  • G.2.4 Integration
  • G.2.5 Taylor Series Expansion
• G.3 Linear Algebra
• G.4 Solving Algebraic Equations
• G.5 Solving Differential Equations

G.1 How to Declare Symbolic Variables and Handle Symbolic Expressions

To declare any variable(s) as a symbolic variable, you should use the command ‘ sym’ or ‘ syms’ as follows:

  
 >a=sym('a'); t=sym('t'); x=sym('x');  
 >syms a x y t % or, equivalently and more efficiently 

Once the variables have been declared as symbolic, they can be used in expressions and as arguments to many functions without being evaluated as numeric:

 >f=x^2/(1+tan(x)^2); 
 >ezplot(f,[-pi pi]) 
 >simplify(cos(x)^2+sin(x)^2) % simplify an expression 
 ans = 1 
 >simplify(cos(x)^2-sin(x)^2) % simplify an expression 
 ans = cos(2*x) 
 >simplify(2*sin(x)*cos(x)) % simple expression 
 ans = sin(2*x) 
 >simplify(sin(x)*cos(y)+cos(x)*sin(y)) % simple expression 
 ans = sin(x+y) 
 >eq1=expand((x+y)^3-(x+y)^2) % expand 
 eq1 = x^3 + 3*x^2*y - x^2 + 3*x*y^2 - 2*x*y + y^3 - y^2 
 >collect(eq1,y) % collect similar terms in descending order w.r.t. y  
 ans = y^3 + (3*x-1)*y^2 + (3*x^2-2*x)*y + x^3 - x^2 
 >factor(eq1) % factorize 
 ans = [ x + y - 1, x + y, x + y] 
 >horner(eq1) % nested multiplication form 
 ans = x*(x*(x + 3*y - 1) - 2*y + 3*y^2) - y^2 + y^3 
 >pretty(ans) % pretty form 
 2 2 3 
 x (x (x ...