The accuracy of Simpson's rule quadrature for a step size h is
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Numerical Methods
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The table below gives values of a function F(x) obtained for values of x at intervals of 0.25.
The value of the integral of the function between the limits 0 to 1 using Simpson's rule is
| x | 0 | 0.25 | 0.5 | 0.75 | 1.0 |
| F(x) | 1 | 0.9412 | 0.8 | 0.64 | 0.50 |
The value of the integral of the function between the limits 0 to 1 using Simpson's rule is
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A numerical solution of the equation f(x) = x + √x - 3 = 0 can be obtained using Newton-Raphson method. If the starting value is x = 2 for the iteration, the value of X that is to be used in the next step is
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To solve the equation 2sin x = x, by Newton Raphson method, the initial guess value is chosen to be x = 2. Consider x in radius only. The value of x (in radius) obtained after one iteration will be closed to
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A 2nd degree polynomial, f(x) has values of 1, 4 and 15 at x = 0, 1 and 2, respectively. The integral is to be estimated by applying the trapezoidal rule to this data. What is the error (defined as "true value - approximate value") in the estimate?
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Starting from x0 = 1, one step of Newton-Raphson method in solving the equation x3 + 3x - 7 = 0 gives the next value (x1) as
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The equation x3 - x2 + 4x - 4 = 0 is to be solved using the Newton-Raphson method. If x = 2 is taken as the initial approximation of the solution, then the next approximation using this method will be
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The iteration step in order to solve for the cube roots of a given number N using the Newton-Raphson's method is
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Equation ex - 1 = 0 is required to be solved using Newton's method with an initial guess x0 = -1. Then, after one step of Newton's method, estimate x1 of the solution will be given by
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Match List-I with List-II and select the correct answer:
| List-I | List-II |
| a. Newton-Raphson method | 1. Solving nonlinear equations |
| b. Runge-Kutta method equations | 2. Solving simultaneous linear equations |
| c. Simpson's Rule equations | 3. Solving ordinary differential |
| d. Gauss elimination | 4. Numerical integration |
| 5. Interpolation | |
| 6. Calculation of Eigenvalues |
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A calculator has accuracy up to 8 digits after decimal place. The value of when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is
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The following table lists an nth order polynomial f(x) = an xn + an - 1 xn - 1 + , .... a1x + a0 and the forward difference evaluated at equally spaced values of x. The order of the polynomial is
| x | f(x) | Δf | Δ2f | Δ3f |
| -0.4 | 1.7648 | -0.2965 | 0.089 | -0.03 |
| -0.3 | 1.4683 | -0.2075 | 0.059 | -0.0228 |
| -0,2 | 1.2608 | -0.1485 | 0.0362 | -0.0156 |
| -0.1 | 1.1123 | -0.1123 | 0.0206 | -0.0084 |
| 0 | 1 | -0.0917 | 0.0122 | -0.0012 |
| 0.1 | 0.9083 | -0.0795 | 0.011 | 0.006 |
| 0.2 | 0.8288 | -0.0685 | 0.017 | 0.0132 |
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P(0, 3), Q(0.5, 4) and R(1, 5) are three points on the curve defined by f(x). Numerical integration is carried out using both Trapezoidal rule and Simpson's rule within limits x = 0 and x = 1 for the curve. The difference between the two results will be
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The recursion relation to solve x = e-x using Newton-Raphson method is
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Given that one root of the equation x3 - 10x2 + 31x - 30 = 0 is 5, the other two roots are
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The bisection method is applied to compute a zero of the function f(x) = x4 - x3 - x2 - 4 in the interval [1, 9]. The method converges to a solution after . . . . . . . . iterations.
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The square root of a number N is to be obtained by applying the Newton Raphson iterations to the equation x2 - N = 0. If i denotes the iteration index, the correct iterative scheme will be
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During the numerical solution of a first order differential equation using Euler method with step size h, the local truncation error is of order of
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With respect to the numerical evaluation of the definite integral where a and b are given, which of the following statements is/are TRUE?
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.
I. The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
II. The value of K obtained using the Simpson's rule is always equal to the exact value of the definite integral.
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The following equation needs to be numerically solved using the Newton-Raphson method. x3 + 4x - 9 = 0. The iterative equation for this purpose is (k indicates the iteration level)
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