Chapter 7, Problem 9P

Use the program developed in Prob. 7.8 to determine the roots of the equations in Prob. 7.5.

To determine

To calculate: Create a program for finding roots of polynomial equation using Bairstow’s method for the functions given below.

(a) $f_3\left(x\right)=0.7x^3-4x^2+6.2x-2$

(b) $f_4\left(x\right)=-3.704x^4+16.3x^2-21.97x+9.34$

(c) $f_4\left(x\right)=x^4-2x^3+6x^2-2x+5$

Answer to Problem 9P

Solution:

The roots for the Problem 7.5(a) are $3.2786,2.0000,\text{ 0}\text{.4357}$.

The roots for the Problem 7.5(b) are $\text{ 2}\text{.2947},\text{ 1}\text{.1525},\text{ 0}\text{.9535}$.

The roots for the Problem 7.5(c) are $\text{ 1}\text{.0000+2}\text{.0000}i,\text{ 1}\text{.0000-2}\text{.0000}i,0.\text{0000+1}\text{.0000}i,$ $0.\text{0000-1}\text{.0000}i$.

Explanation of Solution

Given Information:

Finding the roots of the equation given in Problem 7.5.

Calculation:

Code for Problem 7.5(a):

OptionExplicit

SubPolynomialRoot()

Dim num AsInteger

DimmaxItrAsInteger,ierAsInteger,iAsInteger

Dimarr(10)AsDouble, real(10)AsDouble

Dimimaginary(10)AsDouble' Variable to provide negative or positive sign

Dim r AsDouble, s AsDouble, es AsDouble

'In example, the function is having 3 as highest power, so num is initialized to 3.

num =3

'Values of coefficients of the variables in function is given

arr(0) = -2: arr(1) = 6.2: arr(2) = -4: arr(3) = 0.7

'Maximum number of iterations are defined.

maxItr=20

es =0.0001

r =-1

s =-1

CallBairstow_method(arr(), num, es, r, s,maxItr, real(), imaginary(),ier)

'Loop will run to all the powers of function to assign sign between real and imaginary numbers.

Fori=1To num

' If the value at iteration is greater than or equal to 0, assign '+' sign.

If imaginary(i)>=0Then

MsgBox real(i)&" + "& imaginary(i)&"i"

' If the value at iteration is lesser than 0, assign '-' sign.

Else

MsgBox real(i)&" - "& Abs(imaginary(i))&"i"

EndIf

Nexti

EndSub

'method with arguments to apply bairstow method in function.

SubBairstow_method(arr,nn, es,rr, ss,maxItr, real, imaginary,ier)

DimiterAsInteger, num AsInteger,iAsInteger

Dim r AsDouble, s AsDouble, ea1 AsDouble, ea2 AsDouble

Dim determine AsDouble,drAsDouble, ds AsDouble

Dim r_1 AsDouble, i1 AsDouble, r_2 AsDouble, i2 AsDouble

Dimbrr(10)AsDouble,crr(10)AsDouble

' Assigning values.

r =rr

s = ss

num =nn

ier=0

ea1 =1

ea2 =1

Do

'Loop will get terminated if power is less than 3 or if iteration is equal to or greater than maximum iteration.

If num <3Oriter>=maxItrThenExitDo

iter=0

Do

'Incrementing iter variable by 1.

iter=iter+1

'assigning values of array into another array.

brr(num)=arr(num)

'Disc brr(num-1) is calculated.

brr(num -1)=arr(num -1)+ r *brr(num)

crr(num)=brr(num)

'Disc crr(num-1) is calculated.

crr(num -1)=brr(num -1)+ r *crr(num)

'Loop will run for powers less than 3 till power 0.

Fori= num -2To0Step-1

'Values of brr(i) and crr(i) are updated.

brr(i)=arr(i)+ r *brr(i+1)+ s *brr(i+2)

crr(i)=brr(i)+ r *crr(i+1)+ s *crr(i+2)

Nexti

determine =crr(2)*crr(2)-crr(3)*crr(1)

If determine <>0Then

dr=(-brr(1)*crr(2)+brr(0)*crr(3))/ determine

ds =(-brr(0)*crr(2)+brr(1)*crr(1))/ determine

r = r +dr

s = s + ds

'Values of ea1 and ea2 are determined for different values in loop .

If r <>0Then ea1 =Abs(dr/ r)*100

If s <>0Then ea2 =Abs(ds / s)*100

Else

' Incrementing r and s by 1.

r = r +1

s = s +1

iter=0

EndIf

' If the values of ea1 and ea2 are less than value of es or iterations are completed then loop will be terminated

If ea1 <= es And ea2 <= es Oriter>=maxItrThenExitDo

Loop

'method call to find quadratic roots.

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Formulae to calculate real and imaginary values at the every term of function.

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

num = num -2

'Updating the coeffinents of function at every iteration.

Fori=0To num

arr(i)=brr(i+2)

Nexti

Loop

Ifiter<maxItrThen

If num =2Then

r =-arr(1)/arr(2)

s =-arr(0)/arr(2)

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Use the values of i1, i2, r_1 and r_2 to determine real(num), imaginary(num), real(num-1) and imaginary(num-1).

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

Else

real(num)=-arr(0)/arr(1)

imaginary(num)=0

EndIf

Else

ier=1

EndIf

EndSub

SubQuadroot(r, s, r_1, i1, r_2, i2)

Dim Disc

'Value of Disc is calculated.

Disc = r ^2+4* s

'Condition for Disc>0.

If Disc >0Then

r_1 =(r +Sqr(Disc))/2

r_2 =(r -Sqr(Disc))/2

i1 =0

i2 =0

'Condition if Disc less than or equal to 0.

Else

r_1 = r /2

r_2 = r_1

i1 =Sqr(Abs(Disc))/2

i2 =-i1

EndIf

EndSub

Code for Problem 7.5(b):

OptionExplicit

SubPolynomialRoot()

Dim num AsInteger

DimmaxItrAsInteger,ierAsInteger,iAsInteger

Dimarr(10)AsDouble, real(10)AsDouble

Dimimaginary(10)AsDouble' Variable to provide negative or positive sign

Dim r AsDouble, s AsDouble, es AsDouble

'In example, the function is having 3 as highest power, so num is initialized to 3.

num =3

'Values of coefficients of the variables in function is given

arr(0) = 9.34: arr(1) = -21.97: arr(2) = 16.3: arr(3) = 3.704

'Maximum number of iterations are defined.

maxItr=20

es =0.0001

r =-1

s =-1

CallBairstow_method(arr(), num, es, r, s,maxItr, real(), imaginary(),ier)

'Loop will run to all the powers of function to assign sign between real and imaginary numbers.

Fori=1To num

' If the value at iteration is greater than or equal to 0, assign '+' sign.

If imaginary(i)>=0Then

MsgBox real(i)&" + "& imaginary(i)&"i"

' If the value at iteration is lesser than 0, assign '-' sign.

Else

MsgBox real(i)&" - "& Abs(imaginary(i))&"i"

EndIf

Nexti

EndSub

'method with arguments to apply bairstow method in function.

SubBairstow_method(arr,nn, es,rr, ss,maxItr, real, imaginary,ier)

DimiterAsInteger, num AsInteger,iAsInteger

Dim r AsDouble, s AsDouble, ea1 AsDouble, ea2 AsDouble

Dim determine AsDouble,drAsDouble, ds AsDouble

Dim r_1 AsDouble, i1 AsDouble, r_2 AsDouble, i2 AsDouble

Dimbrr(10)AsDouble,crr(10)AsDouble

' Assigning values.

r =rr

s = ss

num =nn

ier=0

ea1 =1

ea2 =1

Do

'Loop will get terminated if power is less than 3 or if iteration is equal to or greater than maximum iteration.

If num <3Oriter>=maxItrThenExitDo

iter=0

Do

'Incrementing iter variable by 1.

iter=iter+1

'assigning values of array into another array.

brr(num)=arr(num)

'Disc brr(num-1) is calculated.

brr(num -1)=arr(num -1)+ r *brr(num)

crr(num)=brr(num)

'Disc crr(num-1) is calculated.

crr(num -1)=brr(num -1)+ r *crr(num)

'Loop will run for powers less than 3 till power 0.

Fori= num -2To0Step-1

'Values of brr(i) and crr(i) are updated.

brr(i)=arr(i)+ r *brr(i+1)+ s *brr(i+2)

crr(i)=brr(i)+ r *crr(i+1)+ s *crr(i+2)

Nexti

determine =crr(2)*crr(2)-crr(3)*crr(1)

If determine <>0Then

dr=(-brr(1)*crr(2)+brr(0)*crr(3))/ determine

ds =(-brr(0)*crr(2)+brr(1)*crr(1))/ determine

r = r +dr

s = s + ds

'Values of ea1 and ea2 are determined for different values in loop .

If r <>0Then ea1 =Abs(dr/ r)*100

If s <>0Then ea2 =Abs(ds / s)*100

Else

' Incrementing r and s by 1.

r = r +1

s = s +1

iter=0

EndIf

' If the values of ea1 and ea2 are less than value of es or iterations are completed then loop will be terminated

If ea1 <= es And ea2 <= es Oriter>=maxItrThenExitDo

Loop

'method call to find quadratic roots.

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Formulae to calculate real and imaginary values at the every term of function.

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

num = num -2

'Updating the coeffinents of function at every iteration.

Fori=0To num

arr(i)=brr(i+2)

Nexti

Loop

Ifiter<maxItrThen

If num =2Then

r =-arr(1)/arr(2)

s =-arr(0)/arr(2)

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Use the values of i1, i2, r_1 and r_2 to determine real(num), imaginary(num), real(num-1) and imaginary(num-1).

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

Else

real(num)=-arr(0)/arr(1)

imaginary(num)=0

EndIf

Else

ier=1

EndIf

EndSub

SubQuadroot(r, s, r_1, i1, r_2, i2)

Dim Disc

'Value of Disc is calculated.

Disc = r ^2+4* s

'Condition for Disc>0.

If Disc >0Then

r_1 =(r +Sqr(Disc))/2

r_2 =(r -Sqr(Disc))/2

i1 =0

i2 =0

'Condition if Disc less than or equal to 0.

Else

r_1 = r /2

r_2 = r_1

i1 =Sqr(Abs(Disc))/2

i2 =-i1

EndIf

EndSub

Code for Problem 7.5(c):

OptionExplicit

SubPolynomialRoot()

Dim num AsInteger

DimmaxItrAsInteger,ierAsInteger,iAsInteger

Dimarr(10)AsDouble, real(10)AsDouble

Dimimaginary(10)AsDouble' Variable to provide negative or positive sign

Dim r AsDouble, s AsDouble, es AsDouble

'In example, the function is having 4 as highest power, so num is initialized to 4.

num =4

'Values of coefficients of the variables in function is given

arr(0) = 5: arr(1) = -2: arr(2) = 6: arr(3) = -2: arr(4) = 1

'Maximum number of iterations are defined.

maxItr=20

es =0.0001

r =-1

s =-1

CallBairstow_method(arr(), num, es, r, s,maxItr, real(), imaginary(),ier)

'Loop will run to all the powers of function to assign sign between real and imaginary numbers.

Fori=1To num

' If the value at iteration is greater than or equal to 0, assign '+' sign.

If imaginary(i)>=0Then

MsgBox real(i)&" + "& imaginary(i)&"i"

' If the value at iteration is lesser than 0, assign '-' sign.

Else

MsgBox real(i)&" - "& Abs(imaginary(i))&"i"

EndIf

Nexti

EndSub

'method with arguments to apply bairstow method in function.

SubBairstow_method(arr,nn, es,rr, ss,maxItr, real, imaginary,ier)

DimiterAsInteger, num AsInteger,iAsInteger

Dim r AsDouble, s AsDouble, ea1 AsDouble, ea2 AsDouble

Dim determine AsDouble,drAsDouble, ds AsDouble

Dim r_1 AsDouble, i1 AsDouble, r_2 AsDouble, i2 AsDouble

Dimbrr(10)AsDouble,crr(10)AsDouble

' Assigning values.

r =rr

s = ss

num =nn

ier=0

ea1 =1

ea2 =1

Do

'Loop will get terminated if power is less than 3 or if iteration is equal to or greater than maximum iteration.

If num <3Oriter>=maxItrThenExitDo

iter=0

Do

'Incrementing iter variable by 1.

iter=iter+1

'assigning values of array into another array.

brr(num)=arr(num)

'Disc brr(num-1) is calculated.

brr(num -1)=arr(num -1)+ r *brr(num)

crr(num)=brr(num)

'Disc crr(num-1) is calculated.

crr(num -1)=brr(num -1)+ r *crr(num)

'Loop will run for powers less than 3 till power 0.

Fori= num -2To0Step-1

'Values of brr(i) and crr(i) are updated.

brr(i)=arr(i)+ r *brr(i+1)+ s *brr(i+2)

crr(i)=brr(i)+ r *crr(i+1)+ s *crr(i+2)

Nexti

determine =crr(2)*crr(2)-crr(3)*crr(1)

If determine <>0Then

dr=(-brr(1)*crr(2)+brr(0)*crr(3))/ determine

ds =(-brr(0)*crr(2)+brr(1)*crr(1))/ determine

r = r +dr

s = s + ds

'Values of ea1 and ea2 are determined for different values in loop .

If r <>0Then ea1 =Abs(dr/ r)*100

If s <>0Then ea2 =Abs(ds / s)*100

Else

' Incrementing r and s by 1.

r = r +1

s = s +1

iter=0

EndIf

' If the values of ea1 and ea2 are less than value of es or iterations are completed then loop will be terminated

If ea1 <= es And ea2 <= es Oriter>=maxItrThenExitDo

Loop

'method call to find quadratic roots.

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Formulae to calculate real and imaginary values at the every term of function.

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

num = num -2

'Updating the coeffinents of function at every iteration.

Fori=0To num

arr(i)=brr(i+2)

Nexti

Loop

Ifiter<maxItrThen

If num =2Then

r =-arr(1)/arr(2)

s =-arr(0)/arr(2)

CallQuadroot(r, s, r_1, i1, r_2, i2)

'Use the values of i1, i2, r_1 and r_2 to determine real(num), imaginary(num), real(num-1) and imaginary(num-1).

real(num)= r_1

imaginary(num)= i1

real(num -1)= r_2

imaginary(num -1)= i2

Else

real(num)=-arr(0)/arr(1)

imaginary(num)=0

EndIf

Else

ier=1

EndIf

EndSub

SubQuadroot(r, s, r_1, i1, r_2, i2)

Dim Disc

'Value of Disc is calculated.

Disc = r ^2+4* s

'Condition for Disc>0.

If Disc >0Then

r_1 =(r +Sqr(Disc))/2

r_2 =(r -Sqr(Disc))/2

i1 =0

i2 =0

'Condition if Disc less than or equal to 0.

Else

r_1 = r /2

r_2 = r_1

i1 =Sqr(Abs(Disc))/2

i2 =-i1

EndIf

EndSub

Output for Problem 7.5(a):

Now, run the code by pressing ‘F5’ key. Hence the output will be,

1^st Root:

2^nd Root:

3^rd Root:

Output for Problem 7.5(b):

Now, run the code by pressing ‘F5’ key. Hence the output will be,

1^st Root:

2^nd Root:

3^rd Root:

Output for Problem 7.5(c):

Now, run the code by pressing ‘F5’ key. Hence the output will be,

1^st Root:

2^nd Root:

3^rd Root:

4^th Root: