#  MCS471 Trapezoidal and Simpson's Rules Example Maple Worksheet
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# Sample User Defined Function:   
#         Related to Normal Probability Distribution by Scaling:
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> 
f:=x->exp(-x^2);
                                                2
                               f := x -> exp(- x )

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#  Remark:  Maple Numerical Integration Functions  
#           Require Maple Student Calculus Package
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> 
with(student):
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# Maple Student Function Trapezoidal Rule  has format: trapezoid("Function(x)",x="A".."B"[,NumberSubintervals"]); 
# Remark:  The number of subintervals is optional. 
#          Below, "A = 0", "B = 1" and "n = 20" subintervals for above user defined 
#          function.  
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> 
trapezoid(f(x),x=0..1,20);
                           /  19                 \
                           |-----                |
                           | \                 2 |
               1/40 + 1/20 |  )   exp(- 1/400 i )| + 1/40 exp(-1)
                           | /                   |
                           |-----                |
                           \i = 1                /

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# Remark:   See Maple Language Help for more examples. 
#           The "trapezoid" function first the discrete formula.   
#           Maple "evalf" produces 10 digit floating point value.
# Note:   "n = 20" subintervals means "n+1 = 21" points or function evaluations.
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> 
TR21:=evalf(");
                               TR21 := .7466708370

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# Maple Student Function Simpson's (1/3) Rule  has format: simpson("Function(x)",x="A".."B"[,NumberSubintervals"]); 
# Remark:  The number of subintervals is optional. 
#          Below, "A=0", "B=1" and "n=20" subintervals for above user defined 
#          function.
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# CAUTION:  The Maple "simpson" also needs s "with(student):" statement above.
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> 
simpson(f(x),x=0..1,20);
                                      /  10                         \
                                      |-----                        |
                                      | \                         2 |
           1/60 + 1/60 exp(-1) + 1/15 |  )   exp(- (1/10 i - 1/20) )|
                                      | /                           |
                                      |-----                        |
                                      \i = 1                        /

                       /  9                  \
                       |-----                |
                       | \                 2 |
                + 1/30 |  )   exp(- 1/100 i )|
                       | /                   |
                       |-----                |
                       \i = 1                /

> 
SR21:=evalf(");
                               SR21 := .7468241839

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# Remark:  Check Relative Difference of Trapezaoidal to Simpson's Rule:
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> 
RelDiff:=(TR21/SR21-1)*100;
                              RelDiff := -.02053320

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# Remark:  Hence a 0.02% difference of Trapezaoidal to Simpson's Rule:
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# 
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# Remark:  Next, we try 40 subintervals or 41 points (function evaluations).
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> 
TR41:=evalf(trapezoid(f(x),x=0..1,40),25);
                       TR41 := .7467858112389793003907348

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# Remark:  In "evalf" function, 25 digits were requested in the last, optional  
#          argument.
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> 
SR41:=evalf(simpson(f(x),x=0..1,40),25);
> 
RelDiff41:=(TR41/SR41-1)*100;
                       SR41 := .7468241360053479327763032

                             RelDiff41 := -.00513170

# Remark:  Now the Relative Difference is 0.005%, 
#          consistent with ratio of Simpson's to trapezoidal theoretical errors 
#          of O(h^2) which becomes O(h^2/4) upon bisecting the step size "h".