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CHAPTER 2
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
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
start
sum = 0
count = 0
INPUT
value
value =
“end of data”
value =
“end of data”
sum = sum + value
count = count + 1
T
F
count > 0
average = sum/count
end
T
F
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, i
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, i
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 = b2 - 4.ac
IF (d >= 0) THEN
r1 = (-b + SQRT(d))/(2*a)
r2 = (-b - SQRT(d))/(2*a)
ELSE
r1 = -b/(2*a)
r2 = r
i1 = SQRT(ABS(d))/(2*a)
i2 = -i
END IF
END IF
END
The answers for the 3 test cases are: ( a ) −0.438, -4.56; ( b ) 0.5; ( c ) −1.25 + 2.33 i ; −1.25 −
2.33 i.
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.
start
INPUT
x, n
i > n
end
i = 1
true = sin(x)
approx = 0
factor = 1
approx approx
x
factor
i
i -
2 1
error
true approx
true
OUTPUT
i,approx,error
i = i + 1
F
T
factor=factor(2i-2)(2i-1)
Pseudocode:
SUBROUTINE Sincomp(n,x)
i = 1
true = SIN(x)
approx = 0
factor = 1
DO
IF i > n EXIT
approx = approx + (-1)
i-
2
i-
/ 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.
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 * (2i-2) * (2i-1)
END DO
END
OUTPUT:
order true approx error
4 0.9974950 0.9973912 1.0403229E-
5 0.9974950 0.9974971 -2.1511559E-
6 0.9974950 0.9974950 0.0000000E+
7 0.9974950 0.9974951 -1.1950866E-
8 0.9974950 0.9974949 1.1950866E-
9 0.9974950 0.9974915 3.5255053E-
10 0.9974950 0.9974713 2.3782223E-
11 0.9974950 0.9974671 2.7965026E-
12 0.9974950 0.9974541 4.0991469E-
13 0.9974950 0.9974663 2.8801586E-
14 0.9974950 0.9974280 6.7163869E-
15 0.9974950 0.9973251 1.7035959E-
Press any key to continue
The errors can be plotted versus the number of terms:
1.E-
1.E-
1.E-
1.E-
1.E-
1.E+
1.E+
1.E+
0 5 10 15
error
n F
2.10 Programs vary, but results are
Bismarck = −10.842 t = 0 to 59
Yuma = 33.040 t = 180 to 242
n A
Step v (12)
t
Error is halved when step is halved
Fortran 90 VBA
Subroutine BubbleFor(n, b)
Implicit None
!sorts an array in ascending
!order using the bubble sort
Integer(4)::m, i, n
Logical::switch
Real::a(n),b(n),dum
m = n - 1
Do
switch = .False.
Do i = 1, m
If (b(i) > b(i + 1)) Then
dum = b(i)
b(i) = b(i + 1)
b(i + 1) = dum
switch = .True.
End If
End Do
If (switch == .False.) Exit
m = m - 1
End Do
End
Option Explicit
Sub Bubble(n, b)
'sorts an array in ascending
'order using the bubble sort
Dim m As Integer, i As Integer
Dim switch As Boolean
Dim dum As Single
m = n - 1
Do
switch = False
For i = 1 To m
If b(i) > b(i + 1) Then
dum = b(i)
b(i) = b(i + 1)
b(i + 1) = dum
switch = True
End If
Next i
If switch = False Then Exit Do
m = m - 1
Loop
End Sub
2.14 Here is a flowchart for the algorithm:
Function Vol(R, d)
pi = 3.
d < R
d < 3 * R
V1 = pi * R^3 / 3
V2 = pi * R^2 (d – R)
Vol = V1 + V
Vol =
“Overtop”
End Function
Vol = pi * d^3 / 3
Here is a program in VBA:
Option Explicit
Function Vol(R, d)
Dim th As Single, r As Single
Const pi As Single = 3.
r = Sqr(x ^ 2 + y ^ 2)
If x < 0 Then
If y > 0 Then
th = Atn(y / x) + pi
ElseIf y < 0 Then
th = Atn(y / x) - pi
Else
th = pi
End If
Else
If y > 0 Then
th = pi / 2
ElseIf y < 0 Then
th = -pi / 2
Else
th = 0
End If
End If
Polar = th * 180 / pi
End Function
The results are:
x y θ
4.18 f(x) = x-1-1/2*sin(x)
f '(x) = 1-1/2*cos(x)
f ''(x) = 1/2*sin(x)
f '''(x) = 1/2*cos(x)
f
IV
(x) = -1/2*sin(x)
Using the Taylor Series Expansion (Equation 4.5 in the book), we obtain the following
st
nd
rd
, and 4
th
Order Taylor Series functions shown below in the Matlab program-
f1, f2, f4. Note the 2
nd
and 3
rd
Order Taylor Series functions are the same.
From the plots below, we see that the answer is the 4
th
Order Taylor Series expansion.
x=0:0.001:3.2;
f=x-1-0.5*sin(x) ;
subplot(2,2,1);
plot(x,f);grid;title('f(x)=x-1-0.5*sin(x)');hold on
f1=x-1.5 ;
e1=abs(f-f1); %Calculates the absolute value of the
difference/error
subplot(2,2,2);
plot(x,e1);grid;title('1st Order Taylor Series Error');
f2=x-1.5+0.25.((x-0.5pi).^2);
e2=abs(f-f2);
subplot(2,2,3);
plot(x,e2);grid;title('2nd/3rd Order Taylor Series Error');
f4=x-1.5+0.25.((x-0.5pi).^2)-(1/48)((x-0.5pi).^4);
e4=abs(f4-f);
subplot(2,2,4);
plot(x,e4);grid;title('4th Order Taylor Series Error');hold off
0 1 2 3 4
0
1
2
3
f(x )= x -1-0.5*s in(x )
0 1 2 3 4
0
1s t O rder Tay lor S eries E rror
0 1 2 3 4
0
2nd/3rd O rder Tay lor S eries E rror
0 1 2 3 4
0
4th O rder Tay lor S eries E rror
4.19 EXCEL WORKSHEET AND PLOTS
First Derivative Approximations Compared to Theoretical
-4.
-2.
-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.
x-values
f'(x)
Theoretical
Backward
Centered
Forward
