Saturday, April 16, 2016

Mathematics: Numerical Analysis: CSIR-NET Previous Year Solved problems.

1.Let f(x)=x+3f(x)=x+3 for x1x1 consider the iteration xn+1=f(xn),x0=0;n0xn+1=f(xn),x0=0;n0.
The possible limits of the iterations are:
  • -1
  • 3
  • 0
  • 3+3+3+3+3+3+
Solution: 
It is given that,
xn+1=f(xn),x0=0;n0xn+1=f(xn),x0=0;n0.
So,
x1=f(x0)=3x1=f(x0)=3
x2=f(x1)=3+3=3+3x2=f(x1)=3+3=3+3
x3=f(x2)=3+3+3x3=f(x2)=3+3+3
Proceeding in same way..
we get limit:
3+3+3+3+3+3+
So, 4th option is correct.

2. The iteration xn+1=12(xn+2xn),n0xn+1=12(xn+2xn),n0 for a given x00x00 is an instance of
  • Fixed point iteration for f(x)=x22f(x)=x22
  • Newton's method for f(x)f(x)
  • Fixed point iteration for f(x)=x2+22xf(x)=x2+22x
  • Newton's method for f(x)=x2+2f(x)=x2+2
Solution:
Recall,
Fixed point iteration method,
The first step in this method is to rewrite the equation f(x)=0f(x)=0 as x=g(x)x=g(x)
starting from  x=x0x=x0 we find the the first approximation on x1x1 as x1=g(x0)x1=g(x0).
then we find x2=g(x1)x2=g(x1)
we can see here, similar situation is here,
comparing with x=g(x)x=g(x)
we get,
g(x)=12×(x+2x)g(x)=12×(x+2x)

So, 3rd option is correct.
Now Recall, newton's method,
xn+1=xnf(xn)f(xn)xn+1=xnf(xn)f(xn)
so, applying this on x22x22 we get the same form described in the question.
So, 2nd option is correct.


3. The following numerical integration formula is exact for all polynomials of degree less than or
equal to 3.
  • Trapezoidal Rule
  • Simpson's 1313rd rule
  • Simpson's 3838rd rule
  • Gauss -Legendre 2 point formula
Solution:
option 2 is correct:
The truncation error for the Simpson method is proportional to the f4(x)f4(x). If the integrand is cubic, then the fourth order derivative is zero. So the quadrature will be exact for this case.
Option 3 is correct:
This method is more accurate than previous.
Option 4 is correct:
Error for this method is:

(b1)2n+1(n!)4(2n+1)[(2n)!]3f(2n)(ξ)(b1)2n+1(n!)4(2n+1)[(2n)!]3f(2n)(ξ)          a<ξ<ba<ξ<b

So, Here n=2n=2, so, Sice fourth derivative is zero, so this method is exact for cubic polynomial.

4. Let f:RRf:RR be a smooth function with non-vanishing derivative. the Newton's method's for finding a root of f(x)=0f(x)=0 is same as
  • fixed point iteration for the map g(x)=xf(x)/f(x)g(x)=xf(x)/f(x)
  • Forward Euler method with unit step length for the differential equation dydx+f(y)f(y)=0dydx+f(y)f(y)=0
  • Fixed point iteration for g(x)=x+f(x)g(x)=x+f(x)
  • Fixed point iteration for g(x)=xf(x)g(x)=xf(x)
Solution:
Newton's method is:
xn+1=xnf(x)f(x)xn+1=xnf(x)f(x)
So fixed point iteration will be the same.
So, 1st option is correct.
Euler's method:
For,
dydx=f(x,y)dydx=f(x,y)
we have,
yn+1=yn+h(f(xn,yn))yn+1=yn+h(f(xn,yn))

here h=1h=1
So, we get,
yn+1=yn+(f(xn,yn))yn+1=yn+(f(xn,yn))
here f(x,y)=f(y)f(y)=0f(x,y)=f(y)f(y)=0
yn+1=yn+1×(f(y)(f(y))yn+1=yn+1×(f(y)(f(y))
So, 2 option is also correct.

5. Consider the function f(x)=2+xf(x)=2+x for x1x1 and the iteration xn+1=f(xn);n0xn+1=f(xn);n0 for x0=1x0=1 what are the possible limits of the iteration?
  • -1
  • 2
  • 1
  • 2+2+2+2+2+2+
Solution:
It is given that,
xn+1=f(xn),x0=1;n0xn+1=f(xn),x0=1;n0.
So,
x1=f(x0)=3x1=f(x0)=3
x2=f(x1)=3+2=2+3x2=f(x1)=3+2=2+3
x3=f(x2)=2+2+3x3=f(x2)=2+2+3
Proceeding in same way..
we get limit:
2+2+2+2+2+2+
4th option is correct.
Note: In CSIR official Key 22 is also an answer.(I don't know how!) 
6.  The iteration xn+1=12(xn+2xn),n0xn+1=12(xn+2xn),n0 for a given x00x00 is an instance of
  • xnxn converges to 22 with rate of convergence 11.
  • xnxn converges to 22 with rate of convergence 22
  • Fixed point iteration for f(x)=x22f(x)=x22
  • Newton's method for f(x)=x22f(x)=x22
Solution:
See question number 22.
4th option is correct.
xnxn converges to 22 , It is evident for the function f(x)=x22f(x)=x22
Newton Raphson Method is said to have quadratic convergence.
So, 2nd option is correct.

7. which of the following approximations for estimating the derivative of a smooth function ff at a point xx is of order 22. (that is the error term is O(h2O(h2.
  • f(x)f(x+h)f(x)hf(x)f(x+h)f(x)h
  • f(x)f(x+h)f(xh)2hf(x)f(x+h)f(xh)2h
  • f(x)3f(x)4f(xh)+f(x2h)2hf(x)3f(x)4f(xh)+f(x2h)2h
  • f(x)3f(x)+4f(x+h)f(x+2h)hf(x)3f(x)+4f(x+h)f(x+2h)h
solution:

Using forward difference formula, we get
hfk=12(fk+2+4fk+13fk)hfk=12(fk+2+4fk+13fk)
f(x)=12h(3f(x)+4f(x+h)f(x+2h))f(x)=12h(3f(x)+4f(x+h)f(x+2h))
So, 44 is correct option.


8.Let fC3([x1,x1])fC3([x1,x1]) where x1=x0h,x1=x0+hx1=x0h,x1=x0+hwith h0h0f(x0)=f0,f(xj)=fjf(x0)=f0,f(xj)=fj for j=1,1j=1,1 and f(x0)=f0f(x0)=f0
then for some ϵ(x1,x1)ϵ(x1,x1) we have,
  • f0=f1f0hh22f(ϵ)f0=f1f0hh22f′′′(ϵ)
  • f0=f1f1hh33f(ϵ)f0=f1f1hh33f′′′(ϵ)
  • f0=f1f12hh26f(ϵ)f0=f1f12hh26f′′′(ϵ)
  • f0=f1f12hh36f(ϵ)f0=f1f12hh36f′′′(ϵ)

Solution:
This question is similar to above question.
3rd option is correct.

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5 comments :

  1. Can u suggest me some good books in numerical analysis?

    ReplyDelete
  2. I am preparing for net exam for upcoming net exam . Very useful information on your blog. Thanks a lot.

    ReplyDelete
  3. Q5 as you have mentioned has two correct answers. Let 4th option be x.
    x=\sqrt{2+\sqrt{2+\sqrt{2+....
    Then x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+....
    or x^2=2+x. Solve this quadratic equation and note that x by definition is >1. Thus x=2.
    Alternatively, since it is a fixed point interation, solution/limit should satisfy x=f(x). That is x=\sqrt{2+x}. Put x=2 and it satisfies.

    ReplyDelete
  4. If we solve Q 5 then we can get the answer Y=2 also

    ReplyDelete
  5. 5th sum option 2 is true. For, suppose that the limit of option 4 is x then x=√2+√2+...
    That is x^2=2+x. Then, positive solution to this quadratic equation is the convergent point of option 4. Hence 2 is!

    ReplyDelete