Numerical Methods Questions
1 f(x) = x3 – 2x – 5
a) Show that there is a root β of f(x) = 0 in the interval [2,3]. The root β is to be estimated using the iterative formula ,2 5 2 0 2 1 1 = =++ x x x n n b) Calculate the values of x1, x2, x3, and x4, giving your answers to 4 sig fig. c) Prove that, to 5 significant figures, β is 2.0946
2 Use the iterative formula
n n cox x x − =+ 1 1 With x1 = 0.5 to find the limit of the sequence x1, x2, x3,……. Correct to 2 decimal places. 3 Starting with x0 = 1, use the iterative formula
( ) 5lnln1 =++ xx n
To find, to 2 decimal places, the value of x1, x2, x3, and x4. 4 The equation 2x = x3 has two roots. Show that the intervals are [1,2] and [9,10]. 5 f( x ) = x 3 – 2 –
, x ≠ 0.
(a) Show that the equation f( x ) = 0 has a root between 1 and 2. An approximation for this root is found using the iteration formula x n + 1 =
nx 1 2
+ , with 0 x = 1.5.
(b) By calculating the values of x 1, x 2, x 3 and x 4, find an approximation to this root, giving your answer to 3 decimal places. (c) By considering the change of sign of f( x ) in a suitable interval, verify that your answer to part (b) is correct to 3 decimal places.
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection (YBC 7289), which gives a sexagesimal numerical approximation of , the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction. Numerical analysis continues this long tradition of practical mathematical calculations. Much like the Babylonian approximation of , modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors. Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology. Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead. These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.
The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following: Advanced numerical methods are essential in making numerical weather prediction feasible. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations. Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically. Hedge funds (private investment funds) use tools from all fields of numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants. Airlines use sophisticated optimization algorithms to decide ticket prices,...
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