State Newton's forward and backward interpolating formula. 15. Using Lagranges find y at x = 2 for the following. X : 0
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NUMERICAL METHIODS Unit I : Solution of equations and eigen value problems
Part B
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Part A 1. If g(x) is continuous in [a,b] then under what condition the iterative method x = g(x) has unique solution in [a,b]. 2. By Newton’s method find an iterative formula to find 1/N. 3. Find the positive root of x3 + 5x – 3 = o usind fixed point iteration method with 0.6 as first approximation. ⎛1 3 ⎞ ⎟⎟ by Gauss – Jordan method. 4. Find inverse of A = ⎜⎜ ⎝2 7⎠ 5. State the convergence and order of convergence for method of false position. 6. Why Gauss Seidel iteration is a method of successive corrections. 7. Compare Gauss Jacobi and Gauss Siedel methods for solving linear system of the form AX = B. 8. State the conditions for convergence of Gauss Siedel method for solving a system of equations. 9. Find an iterative formula to find N where N is a positive number. 10. Compare Gaussian elimination method and Gauss-Jordan method. 11. What type of eigen value can be obtained using power method. 12. State the order of convergence and convergence condition for Newton Raphson,s method. ⎡1 2 ⎤ 13. Find the dominant eigen value of A = ⎢ ⎥ by power method. ⎣3 4⎦ 14. How is the numerically smallest eigen value of A obtained. 15. State two difference between direct and iterative methods for solving system of equations.
1. Consider the non – linear system x2 – 2x – y +0.5 =0 and x2+4y2 – 4 = 0. Use Newton – Raphson method with the starting value (x0, y0) = (2.00, 0.25) and compute (x1, y1), (x2, y2) and (x3, y3) .
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2. Fins the real root of xex – 3 = 0 by regula - falsi method. 3. Find the real root using method of false position for x3 – 2x – 5 = 0 coreect to three decimal places.
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⎡ 2 −1 0 ⎤ 4. Find all the eigen values of the matrix ⎢⎢− 1 2 − 1⎥⎥ by power method (Apply ⎢⎣ 0 − 1 2 ⎥⎦ only 3 iterations).
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5. Use Newton’s backward difference formula to construct an interpolating polynomial of degree 3 for the data: f( - 0.75) = - 0.0718125, f( - 0.5) = - 0.02475, f( - 0.25) = - 0.3349375 and f(0) = 1 1.101. Hence find f (- ). 3 6. Solve the system of equations using Gauss Seidel iterative methods. 20x – y – 2z = 17, 3x + 20y – z = -18, 2x – 3y +20z = 25.
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7.Find the largest eigen values and its corresponding vector of the matrix ⎡ 1 3 − 1⎤ ⎢ 3 2 4 ⎥ by power method. ⎢ ⎥ ⎢⎣− 1 4 10 ⎥⎦
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⎡2 2 3⎤ 8. Using Gauss- Jordan obtain the inverse of the matrix ⎢⎢2 1 1⎥⎥ ⎢⎣1 3 5⎥⎦
9. Using Gauss Seidel method solve the system of equations starting with the values x = 1 , y = -2 and z = 3, x + 3y + 5z = 173.61, x – 27y + 2z = 71.31, 41x – 2y + 3z = 65.46
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10. Solve the following equations by Jacobi’s iteration method x + y + z = 9, 2x – 3y + 4z = 13, 3x + 4y + 5z = 40.
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Unit II : Interpolation and Approxiamtion Part A
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1. Construct a linear interpolating polynomial given the points (x0,y0) and (x1,y1). 2. Obtain the interpolation quadratic polynomial for the given data by using Newton’s forward difference formula. X: 0 2 4 6 Y : -3 5 21 45 3. Obtain the divided difference table for the following data. X : -1 0 2 3 Y : -8 3 1 12 4. Find the polynomial which takes the following values. X : 0 1 2 Y : 1 2 1 5. Define forward, backward, central differences and divided differences. 6. Evaluate ∆10 (1-x) (1-2x) (1-3x)--------(1-10x), by taking h=1. 7. Show that the divided difference operator ∆ is linear. 8. State the order of convergence of cubic spline. 9. What are the natural or free conditions in cubic spline. 10. Find the cubic spline for the following data X:0 2 4 6 Y:1 9 21 41 11. State the properties of divided differences. 1 −1 12. Show that ∆3 ( ) = . bcd a abcd 13. Find the divided differences of f(x) = x3 + x + 2 for the arguments 1,3,6,11. 14. State Newton’s forward and backward interpolating formula. 15. Using Lagranges find y at x = 2 for the following X:0 1 3 4 5 Y:0 1 81 256 625
Part B
1. Using Lagranges interpolation formula find y(10) given that y(5) = 12, y(6) = 13, y(9) = 14 and y(11) = 16.
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2. Find the missing term in the following table x:0 1 2 3 4 y : 1 3 9 - 81 3. From the data given below find the number of students whose weight is between 60 to 70. Wt (x) : 0-40 40-60 60-80 80-100 100-120
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No of students : 250
120
100
70
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4. From the following table find y(1.5) and y’(1) using cubic spline. X : 1 2 3 Y : -8 -1 18
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5. Given sin 450 = 0.7071, sin 500 = 0.7660, sin 550 = 0.8192, sin 600 = 0.8660, find sin 520 using Newton’s forward interpolating formula. 6. Given log 10 654 = 2.8156, log 10 658 = 2.8182, log 10 659 = 2.8189, log 10 661 = 2.8202, find using Lagrange’s formula the value of log 10 656. 7. Fit a Lagrangian interpolating polynomial y = f(x) and find f(5) x:1 3 4 6 y : -3 0 30 132
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8. Find y(12) using Newton’ forward interpolation formula given x : 10 20 30 40 50 y : 46 66 81 93 101
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9. Obtain the root of f(x) = 0 by Lagrange’s inverse interpolation given that f(30) = -30, f(34) = -13, f(38) = 3, f(42) = 18.
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10. Fit a natural cubic spline for the following data x:0 1 2 3 y:1 4 0 -2 11. Derive Newton’s divided difference formula.
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12. The following data are taken from the steam table: 150 160 170 180 Temp0 c : 140 Pressure : 3.685 4.854 6.502 8.076 10.225 Find the pressure at temperature t = 1420 and at t = 1750 13. Find the sixth term of the sequence 8,12,19,29,42.
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14. From the following table of half yearly premium for policies maturing at different ages, estimate the premium for policies maturing at the age of 46. Age x : 45 50 55 60 65 Premium y : 114.84 96.16 83.32 74.48 68.48 15. Form the divided difference table for the following data x : -2 0 3 5 7 y : -792 108 -72 48 -144
8 -252
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Unit III Differentiation and Integration Part A
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1. What the errors in Trapezoidal and Simpson’s rule. 2. Write Simpson’s 3/8 rule assuming 3n intervals. 1 dx using Gaussian quadrature with two points. 3. Evaluate ∫ 1+ x 4 −1 4. In Numerical integration what should be the number of intervals to apply Trapezoidal, Simpson’s 1/3 and Simpson’s 3/8. 1 x 2 dx 5. Evaluate ∫ using Gaussian three point quadrature formula. 1+ x 4 −1 1
6. State two point Gaussian quadratue formula to evaluate
∫ f ( x)dx .
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7. Using Newton backward difference write the formula for first and second order derivatives at the end value x = x0 upto fourth order. dy d2y and at x = x0 using Newtons forward 8. Write down the expression for dx dx 2 difference formula. 9. State Simpson’s 1/3 and Simpson’s 3/8 formula. Π
10. Using trapezoidal rule evaluate ∫ sin xdx by dividing into six equal parts.
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Part B
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1. Using Newton’s backward difference formula construct an interpolating polynomial of degree three and hence find f(-1/3) given f(-0.75) = - 0.07181250, f(-0.5) = - 0.024750, f(-0.25) = 0.33493750, f(0) = 1.10100. 2. Evaluate
dxdy
∫ ∫ 1+ x + y 2 2
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3. Evaluate I =
dx dy
∫∫ x+ y
by Simpson’s 1/3 rule with ∆x = ∆y = 0.5 where 0
