Numerical methods
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 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 1
, 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 =
3 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.[2]
Numerical analysis continues this long tradition of practical mathematical calculations. Much
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 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 1
, 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 =
3 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.[2]
Numerical analysis continues this long tradition of practical mathematical calculations. Much