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Syeda

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Solving cubic equation...

Syeda 30
solve rootfinding roundoff
March 18 2014
2 4

Hi all,

By solving cubic equation in maple (version 17), I got

restart

``

-0.363700352e-2*y^3-.4041941000*y^2+3.397775673*y-2.377540486 = 0

-0.363700352e-2*y^3-.4041941000*y^2+3.397775673*y-2.377540486 = 0

 (1)

"(->)"

[[y = .7709248124], [y = 7.123944371], [y = -119.0286907]]

 (2)

``

Now I want to find these roots through the formula.

 

I solve it generally in Maple.. 

 

``# Suppose

A*y^3+B*y^2+C*y+E = 0

A*y^3+B*y^2+C*y+E = 0

 (3)

NULL

A := -0.363700352e-2:

B := -.4041941000:

C := 3.397775673:

E := -2.377540486:

``

A*y^3+B*y^2+C*y+E = 0

 

A*y^3+B*y^2+C*y+E = 0

 (4)

``

y1 := (1/6)*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)/A-(2/3)*(3*A*C-B^2)/(A*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3))-(1/3)*B/A

-45.82526955*(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)-36.74197467/(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)-37.04460717

 (5)

"(=)"

-119.0286907-0.1e-8*I

 (6)

y2 := y = -(1/12)*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)/A+(1/3)*(3*A*C-B^2)/(A*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3))-(1/3)*B/A+(1/2*I)*sqrt(3)*((1/6)*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)/A+(2/3)*(3*A*C-B^2)/(A*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)))

y = 22.91263477*(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)+18.37098733/(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)-37.04460717+((1/2)*I)*3^(1/2)*(-45.82526955*(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)+36.74197467/(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3))

 (7)

"(=)"

y = .770924807+0.1772050808e-7*I

 (8)

y3 := y = -(1/12)*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)/A+(1/3)*(3*A*C-B^2)/(A*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3))-(1/3)*B/A-(1/2*I)*sqrt(3)*((1/6)*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)/A+(2/3)*(3*A*C-B^2)/(A*(-108*E*A^2+36*A*B*C+12*sqrt(3)*sqrt(27*A^2*E^2-18*A*B*C*E+4*A*C^3+4*B^3*E-B^2*C^2)*A-8*B^3)^(1/3)))

y = 22.91263477*(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)+18.37098733/(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)-37.04460717-((1/2)*I)*3^(1/2)*(-45.82526955*(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3)+36.74197467/(.7114884222-(0.5542993294e-1*I)*3^(1/2))^(1/3))

 (9)

"(=)"

y = 7.123944373-0.1692050808e-7*I

 (10)

``


y1, y2, y3 formulas are computed by Maple by solving it for general formula.
But, now I got answers in real and imaginery parts, i.e

 

y1 = -119.0286907-1.*10^(-9)*I

y2 = .770924807+1.772050808*10^(-8)*I

y3 = 7.123944373-1.692050808*10^(-8)*I

 

Why, is it so?

 

 

I want answers in simple forum directly only by using these formulas. As i have to show the proof!

Thanks in advance

 

Download qstn.mw

How to find "x" by substituting values i...

Syeda 30
student matrix
January 23 2014
1 3

I have two matrices.How to find matrix "x". Equation of x is given. I can evaluate the value of 'x' at a point. 

 

z[1] = [0.3042939e-2, .135433537, .263300507, .376372832, .474749796, .556813639, .593373102, 0.3250283e-2, .134445703, .260238867]:

z[2] = [.997953273, .988583777, .96233161, .923889683, .877336956, .827642812, .801825939, .994477871, .978659033, .947179739]:
 

eq1 := x = 1.782005573*z[1]^2+2.483310992*z[1]+1.360919422-3.123062564*10^(-9)*sqrt(3.174428983*10^17*z[1]^4+8.847440780*10^17*z[1]^3+1.461867487*10^18*z[1]^2+1.178106706*10^18*z[1]+1.600992583*10^18*z[2]-1.484747647*10^18)

x = 1.782005573*z[1]^2+2.483310992*z[1]+1.360919422-0.3123062564e-8*(0.3174428983e18*z[1]^4+0.8847440780e18*z[1]^3+0.1461867487e19*z[1]^2+0.1178106706e19*z[1]+0.1600992583e19*z[2]-0.1484747647e19)^(1/2)

 (1)

 

Download Find_x.mw

Real roots only...

Syeda 30
real roots
January 17 2014
1 1

I want to find real roots only.  Cannot we find a simplified formula for x in this case which gives only real roots? 

 

 

``

eq1 := a^2*x^3+Typesetting:-delayDotProduct(2*a*b-Typesetting:-delayDotProduct(a^2, e), x^2)+(-2*a*b*c^2-a*c+b^2)*x-c*b-d-b^2*e = 0:

``

# Formula

eq2 := A*x^3+B*x^2+C*x+E = 0:

``

NULL

a := .7438:

b := 15.12*z[1]+10.85*z[1]^2:

c := 18.92-17.76*z[2]:

d := -.9224:

e := 2.106-5.317*z[2]+2.87*z[2]^2:NULL

NULL

A := a^2:

B := -a^2*e+2*a*b:

C := -2*a*b*e^2-a*c+b^2:

E := -b^2*e-b*c-d:

``

eq2

.55323844*x^3+(-1.165120155+2.941568785*z[2]-1.587794323*z[2]^2+22.492512*z[1]+16.140460*z[1]^2)*x^2+(-1.4876*(15.12*z[1]+10.85*z[1]^2)*(2.106-5.317*z[2]+2.87*z[2]^2)^2-14.072696+13.209888*z[2]+(15.12*z[1]+10.85*z[1]^2)^2)*x-(15.12*z[1]+10.85*z[1]^2)^2*(2.106-5.317*z[2]+2.87*z[2]^2)-(15.12*z[1]+10.85*z[1]^2)*(18.92-17.76*z[2])+.9224 = 0

 (1)

``

``# Putting z1 and z2 value

"(->)"

.55323844*x^3+14.11629660*x^2+83.26002702*x-3.52866181 = 0

 (2)

 

"(->)"

[[x = 0.4208050385e-1], [x = -9.354079555], [x = -16.20375615]]

 (3)

``

``

 

Download cubic.mw

Error, attempting to assign to "D" which...

Syeda 30
error student warning
January 17 2014
1 3

While solving cubic formula, i got this warning in Maple.

Why D variable is protected? 

eq1 := A*x^3+B*x^2+C*x+D = 0

``

A := x^2:

B := 2*x*y-4*z^2:

C := -2*x*y*z+x^2-y*z:

D := 4*x^2-x*z

Error, attempting to assign to `D` which is protected.  Try declaring `local D`; see ?protect for details.

  

``

eq1

x^5+(2*x*y-4*z^2)*x^2+(-2*x*y*z+x^2-y*z)*x+D = 0

 (1)

 

 

How to define/set a range in Maple?...

Syeda 30
January 14 2014
3 5

Can we define/set a range in Maple. e.g 


I have the following equation:

y = 1.048 + 1.02*x + 6.118*(z-4.041*x^2) + 16.22*(z^2) +6.241* (x*z)

The value of z is within 0.001 - 0.543, y is from 1 - 12 and x is from 0.001 - 0.7

How should I define it in Maple, so while solving equations it read the values within the given range? 


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