Solutions Manual for Numerical Methods for Engineers (8th Edition) by Chapra and Canale

All 31 Chapters Covered




SOLUTION MANUAL

,CHAPTER 2
2.1
IF x < 10 THEN IF x < 5
THEN
x = 5 ELSE
PRINT x END
IF
ELSE
DO
IF x < 50 EXIT
x = x - 5 END DO
END IF

2.2
Step 1: Start
Step 2: Initialize sum and count to zero Step 3:
Examine top card.
Step 4: If it says “end of data” proceed to step 9; otherwise, proceed to next step. Step 5:
Add value from top card to sum.
Step 6: Increase count by 1.
Step 7: Discard top card Step
8: Return to Step 3.
Step 9: Is the count greater than zero?
If yes, proceed to step 10. If
no, proceed to step 11.
Step 10: Calculate average = sum/count
Step 11: End
2.3
start


sum = 0
count = 0




T
count > 0
INPUT
value
F
average = sum/count

value = T
“end of data”


F end
sum = sum + value
count = count + 1

,
,2.4
Students could implement the subprogram in any number of languages. The following Fortran 90
program is one example. It should be noted that the availability of complex variables in Fortran
90, would allow this subroutine to be made even more concise. However, we did not exploit this
feature, in order to make the code more compatible with Visual BASIC, MATLAB, etc.
PROGRAM Rootfind IMPLICIT NONE
INTEGER::ier
REAL::a, b, c, r1, i1, r2, i2
DATA a,b,c/1.,5.,2./
CALL Roots(a, b, c, ier, r1, i1, r2, i2) IF (ier .EQ. 0) THEN
PRINT *, r1,i1," i"
PRINT *, r2,i2," i" ELSE
PRINT *, "No roots" END IF
END
SUBROUTINE Roots(a, b, c, ier, r1, i1, r2, i2) IMPLICIT NONE
INTEGER::ier
REAL::a, b, c, d, r1, i1, r2, i2 r1=0.
r2=0.
i1=0.
i2=0.
IF (a .EQ. 0.) THEN IF (b <> 0)
THEN
r1 = -c/b ELSE
ier = 1 END
IF
ELSE
d = b**2 - 4.*a*c IF (d >= 0)
THEN
r1 = (-b + SQRT(d))/(2*a)
r2 = (-b - SQRT(d))/(2*a) ELSE
r1 = -b/(2*a) r2 = r1
i1 = SQRT(ABS(d))/(2*a)
i2 = -i1 END IF
END IF END


The answers for the 3 test cases are: (a) 0.438, -4.56; (b) 0.5; (c) 1.25 + 2.33i; 1.25
2.33i.

Several features of this subroutine bear mention:
 The subroutine does not involve input or output. Rather, information is passed in and out via the
arguments. This is often the preferred style, because the I/O is left to the discretion of the
programmer within the calling program.
 Note that an error code is passed (IER = 1) for the case where no roots are possible.

,2.5 The development of the algorithm hinges on recognizing that the series approximation of the sine can be
represented concisely by the summation,
n
x 2i 1
i1 (2i 1)!

where i = the order of the approximation. The following algorithm implements this summation:

Step 1: Start
Step 2: Input value to be evaluated x and maximum order n Step 3:
Set order (i) equal to one
Step 4: Set accumulator for approximation (approx) to zero Step 5:
Set accumulator for factorial product (fact) equal to one Step 6:
Calculate true value of sin(x)
Step 7: If order is greater than n then proceed to step 13 Otherwise,
proceed to next step
Step 8: Calculate the approximation with2i-1the formula
approx approx ( 1) i-1 x
Step 9: Determine the error factor
%error true approx
100%
true
Step 10: Increment the order by one
Step 11: Determine the factorial for the next iteration
factor factor (2 i 2) (2 i 1)
Step 12: Return to step 7
Step 13: End

,2.6
start



INPUT
x, n



i=1
true = sin(x)
approx = 0
factor = 1




T
i>n


F
2 i- 1
x
approx=approx+ (- 1)i - 1
factor


true approx
error  100%
true


OUTPUT
i,approx,error


i=i+1



factor=factor(2i-2)(2i-1)


end


Pseudocode:
SUBROUTINE Sincomp(n,x) i = 1
true = SIN(x) approx = 0
factor = 1 DO
IF i > n EXIT
approx = approx + (-1)i-1 x2 i-1 / factor error = Abs(true - approx) /
true) * 100 PRINT i, true, approx, error
i=i+1
factor = factor (2 i-2) (2 i-1) END DO
END

,2.7 The following Fortran 90 code was developed based on the pseudocode from Prob. 2.6:

PROGRAM Series
IMPLICIT NONE
INTEGER::n REAL::x
n = 15
x = 1.5
CALL Sincomp(n,x) END

SUBROUTINE Sincomp(n,x) IMPLICIT
NONE
INTEGER::n,i,fac
REAL::x,tru,approx,er i = 1
tru = SIN(x) approx = 0.
fac = 1
PRINT *, " order true approx error"
DO
IF (i > n) EXIT
approx = approx + (-1) ** (i-1) * x ** (2*i - 1) / fac er = ABS(tru - approx) / tru) * 100
PRINT *, i, tru, approx, er i = i + 1
fac = fac * (2*i-2) * (2*i-1) END DO
END

OUTPUT:
order true approx error
1 0.9974950 1.500000 -50.37669
2 0.9974950 0.9375000 6.014566
3 0.9974950 1.000781 -0.3294555
4 0.9974950 0.9973912 1.0403229E-02
5 0.9974950 0.9974971 -2.1511559E-04
6 0.9974950 0.9974950 0.0000000E+00
7 0.9974950 0.9974951 -1.1950866E-05
8 0.9974950 0.9974949 1.1950866E-05
9 0.9974950 0.9974915 3.5255053E-04
10 0.9974950 0.9974713 2.3782223E-03
11 0.9974950 0.9974671 2.7965026E-03
12 0.9974950 0.9974541 4.0991469E-03
13 0.9974950 0.9974663 2.8801586E-03
14 0.9974950 0.9974280 6.7163869E-03
15 0.9974950 0.9973251 1.7035959E-02
Press any key to continue

The errors can be plotted versus the number of terms:

1.E+02
1.E+01 error
1.E+00
1.E-01
1.E-02
1.E-03
1.E-04
1.E-05
0 5 10 15

, Interpretation: The absolute percent relative error drops until at n = 6, it actually yields a perfect
result (pure luck!). Beyond, n = 8, the errors starts to grow. This occurs because of round-off error,
which will be discussed in Chap. 3.

2.8 AQ = 442/5 = 88.4 AH =
548/6 = 91.33
without final
30(88.4) + 30(91.33)
AG = 89.8667
30 + 30
with final
30(88.4) + 30(91.33) + 40(91) 30 +
AG = 90.32
30
The following pseudocode provides an algorithm to program this problem. Notice that the input of
the quizzes and homeworks is done with logical loops that terminate when the user enters a negative
grade:

INPUT number, name INPUT
WQ, WH, WF
nq = 0
sumq = 0 DO
INPUT quiz (enter negative to signal end of quizzes)
IF quiz < 0 EXIT nq = nq + 1
sumq = sumq + quiz END DO
AQ = sumq / nq PRINT AQ
nh = 0
sumh = 0
PRINT "homeworks" DO
INPUT homework (enter negative to signal end of homeworks)
IF homework < 0 EXIT nh = nh +
1
sumh = sumh + homework END DO
AH = sumh / nh
PRINT "Is there a final grade (y or n)" INPUT answer
IF answer = "y" THEN INPUT FE
AG = (WQ * AQ + WH * AH + WF * FE) / (WQ + WH + WF) ELSE
AG = (WQ * AQ + WH * AH) / (WQ + WH) END IF
PRINT number, name$, AG END
2.9

, n F
0 $100,000.00
1 $108,000.00
2 $116,640.00
3 $125,971.20
4 $136,048.90
5 $146,932.81


24 $634,118.07
25 $684,847.52

2.10 Programs vary, but results are

Bismarck = 10.842 t = 0 to 59
Yuma = 33.040 t = 180 to 242

2.11
n A
1 40,250.00
2 21,529.07
3 15,329.19
4 12,259.29
5 10,441.04
2.12

Step v(12) t (%)
2 49.96 -5.2
1 48.70 -2.6
0.5 48.09 -1.3
Error is halved when step is halved
2.13

Fortran 90 VBA

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