Page 212 - Engineering Mathematics Workbook_Final

Numerical Methods

8. The method 3
+
(a) x n+ 1 = − 16 6x +
n
1 x n
y + = (k + 3k ),n = 0,1,.....
1 y +
n n 1 2
4
+
(b) x n+ 1 = 3 2x
n
( , y
k = hf x )
1 n n 3
(c) x =
 2h 2  n+ 1 x − 2

k = hf x + , y + k 1    is used n

n
2
n
 3 3  2
to solve the initial value problem (d) x n+ 1 = x − 2
n
)
' y = f ( , x y = − 10y , ( ) 0 = 2
1
y
12. While solving the equation
The method will produce stable x − 3x + = using the Newton-
2
1 0
results if the step size h satisfies
Raphson method with the initial


(a) 0.2 h  0.5 (b) 0 h  0.5 guess of a root as 1, the value of the
root after one iteration is


(c) 0 h  1 (d) 0 h  0.2
(a) 1.5 (b) 1
6
x
9. The equation x − − =
1 0 has
(c) 0.5 (d) 0
(a) no positive real roots
13. Consider the system of equations
(b) exactly one positive real root  5 2 1   x   13 
     1    




(c) exactly two positive real roots  − 2 5 2     x  2  = − 22



 − 1 2 8     x   14 



(d) all positive real roots    3   
10. The smallest degree of the With the initial guess of the solution

0
, ,x
polynomial that interpolates the data    x x 2 0 3  0  = 1,1,1 , the
1
x -2 -1 0 1 2 3 approximate value of the solution
1
x
f(x) -58 -21 -12 -13 -6 27    x 1 , ,x 1 3  after one iteration by the
2
1
Is
Gauss-Seidel method is
(a) 3 (b) 4


−
−
−
(a) 2, 4.4,1.625 (b) 2, 4, 3
(c) 5 (d) 6


−
(c) 2,4.4,1.625 (d) 2, 4,3
11. Suppose that x is sufficiently close
0
to 3. Which of the following 14. Using Euler’s method taking step size
iterations x n+ 1 = g x n = 0.1, the approximate value of y
( ) will converge
3
to the fixed point x = ? obtained corresponding to x = 0.2

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