TEST BANK FOR Mathematics for Physical Science and Engineering Symbolic Computing

1.1 COMPUTING: HISTORICAL NOTE
1.2 BASICS OF SYMBOLIC COMPUTING
Exercises
Carry out these exercises using the symbolic computation system of your choice.
1.2.1. Reduce the following polynomial in x and z to fully factored form.
3x2 + 3x4 − 4xz + 3x2z + 2x3z + 6x5z + z2 − 4xz2 − 7x2z2 + 6x3z2
− 8x4z2 + z3 + 2xz3 − 5x2z3 + 2x3z3 − 2xz4 + 6x3z4 + z5 − 8x2z5 + 2xz6
Solution:
In maple,
> ZZ := 3*x^2+3*x^4-4*x*z+3*x^2*z+2*x^3*z+6*x^5*z+z^2-4*x*z^2
> -7*x^2*z^2+6*x^3*z^2-8*x^4*z^2+z^3+2*x*z^3-5*x^2*z^3
> +2*x^3*z^3-2*x*z^4+6*x^3*z^4+z^5-8*x^2*z^5+2*x*z^6;
> factor(ZZ):
In mathematica,
ZZ = 3*x^2+3*x^4-4*x*z+3*x^2*z+2*x^3*z+6*x^5*z+z^2-4*x*z^2
-7*x^2*z^2+6*x^3*z^2-8*x^4*z^2+z^3+2*x*z^3-5*x^2*z^3
+2*x^3*z^3-2*x*z^4+6*x^3*z^4+z^5-8*x^2*z^5+2*x*z^6:
Factor[ZZ];
Both symbolic systems give an answer equivalent to
(x − z)(3x − z)(2xz + 1)(x2 + z3 + z + 1)
1.2.2. Because the polynomial in Exercise 1.2.1 can be factored, symbolic computer systems can
easily find the values of x that are its roots. Calling the polynomial in its original form poly,
use one of the following:
3
4 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING
solve(poly=0,x) or Solve[poly==0,x]
Note. Less convenient forms than that found here are usually obtained as the solutions to
more general root-finding problems.
Solution:
Letting ZZ be as defined in the solution to Exercise 1.2.1, execute one of
> solve(ZZ = 0, x);
1
3
z, − 1
2z
, z ,
√
−z3 − z − 1 , −
√
−z3 − z − 1
Solve[ZZ == 0, x]
{ {
x → − 1
2z
}
,
{
x → z
3
}
, {x → z},
{
x → −
√
−1 − z − z3
}
,
{
x →
√
−1 − z − z3
}}
1.2.3. (a) Starting with f(x) = 1+2x+5x2 −3x3, obtain an expansion of f(x) in powers of x+1
by carrying out the following steps: (1) Define a variable s and assign to it the value of
the polynomial representing f(x); (2) Define x to have the value z − 1 and recompute
s; (3) Expand s in powers of z; (4) Define z to have the value x + 1 and recompute
s. You will need to be careful to have x and/or z undefined at various points in this
process.
(b) Expand your final expression for s and verify that it is correct.
Note. Your algebra system may combine the linear and constant terms in a way that causes
(x + 1) not to be explicitly visible.
Solution:
In maple, these steps correspond to this coding:
> s := 1+2*x+5*x^2-3*x^3:
> x:=z-1: s:
> ss:=expand(s):
> x:='x': z:=x+1: ss:
In mathematica,
s = 1 + 2*x + 5*x^2 - 3*x^3;
x = z - 1; s;
ss = Expand[s];
Clear[x]; z = x + 1; ss;
Both symbolic systems give a result equivalent to
7 − 17(x + 1) + 14(x + 1)2 − 3(x + 1)3
1.2.4. Verify the trigonometric identity sin(x + y) = sin x cos y + cos x sin y by simplifying sin(x +
y) − sin x cos y − cos x sin y to the final result zero. Use the commands for simplifying and
expanding (try both even if the first works). These two commands do not always give the
same results.
Solution:
Both work in maple, Expand does not simplify in mathematica.
1.2. BASICS OF SYMBOLIC COMPUTING 5
1.2.5. Obtain numerical values, precise to 30 decimal digits, for π2 and sin 0.1π.
Note. Obtain these 30-digit results even if your computer does not normally compute at this
precision.
Solution:
Execute one of the following (illustrated only for sin 0.1π):
evalf(sin(0.1*Pi), 30);
N[Sin[0.1*Pi], 30]
Both give sin 0.1π = 0.30901 69943 74947 42410 22934 17183.
The result for π2 is 9.86960 44010 89358 61883 44909 9988.
1.2.6. (a) Verify that your computer system knows the special function corresponding to the
following integral:
erf(x) =
√2
π
∫ x
0
e
−t2
dt .
In maple, it is erf(x); in mathematica, Erf[x].
(b) Write symbolic code to evaluate the integral defining erf(x), and check your code by
comparing its output with calls to the function in your symbolic algebra system. Check
values for x = 0, x = 1, and x = ∞.
Note. Infinity is infinity in maple, Infinity in mathematica.
Solution:
erf(0) = 0, erf(1) = 0.842701, erf(∞) = 1.
evalf(2/sqrt(Pi)*int(exp(-t^2),t=0 .. 1));
N[2/Sqrt[Pi]*Integrate[E^(-t^2), {t, 0, 1}]]
1.2.7. Plot the function defined in Exercise 1.2.6 for various ranges of x. Use enough ranges to get
some experience in using your algebra system’s plotting utility.
Solution:
We illustrate for the range (0 · · · 2):
plot(erf(x), x = 0 .. 2);
Plot[Erf[x], {x, 0, 2}]
1.2.8. Obtain the value of erf(π) to 30 decimal places and print out the message "erf(pi)= · · · ".
Note. The value of erf must be supplied directly by the computer; you should not obtain an
answer by typing it in.
Solution:
In maple:
VAL := evalf(erf(Pi),30):
print("erf(pi) = ",VAL);
“erf(pi) = ”, 0.999991123853632358394731620781 or
6 CHAPTER 1. COMPUTERS, SCIENCE, AND ENGINEERING
printf("erf(pi) = %33.30f\n",VAL);