If the error term of the method can be written as a power series in h, then by repeating the extrapolation procedure a number of times, we can obtain methods of higher orders. The method 4. The extrapolation table is given below.
Table 4. Extrapolation table for 4. However, the method which approximates f r x con- tains hr in the denominator. As h is successively decreased to small values, the truncation error decreases, but the roundoff error in the method may increase as we are dividing by a smaller number.
It may happen that after a certain critical value of h, the roundoff error may become more dominant than the truncation error and the numerical results obtained may start worsening as h is further reduced. When f x is given in tabular form, these values may not themselves be exact. If we use the criterion [4. Since in 4. Thus, the method of the form 4. If some of the nodes are known in advance, the order will be reduced.
These methods are known as Newton- Cotes integration methods and have the order n. When both the end points of the interval of integration are used as nodes in the methods, the methods are called closed type methods, otherwise, they are called open type methods.
The error term is given as 4. The error term is given by 4. For large n, some of the weights become negative. This may cause loss of significant digits due to mutual cancellation. Weights of Newton-Cotes Integration Rule 4. The proof is given below. Observe that l 2j x is a polynomial of degree less than or equal to 2n. The methods 4. Nodes and Weights for Lobatto Integration Method 4. Differentiation and Integration The nodes and the corresponding weights for the method 4. Nodes and Weights for Radau Integration Method 4.
The methods of the form 4. Differentiation and Integration Table 4. Methods of the form 4. The nodes and the weights for the integration method 4. We divide the interval [a, b] or [— 1, 1] into a number of subintervals and evaluate the integral in each subinterval by a particular method. The errors in the composite trapezoidal rule 4. This integral can be evaluated numerically by two successive integrations in x any y 4.
Trapezoidal rule If we evaluate the inner integral in 4. Using the trapezoidal rule again in 4. The computational molecule of the method 4. Find the error term. Stockholm Univ. Give the error estimate the values in the table are correctly rounded. Extrapolation Table h O h2 O h4 O h6 0. Royal Inst.
Newton-Cotes Methods 4. How many decimals would be required in function values? JK dx 0. Improve the results using Romberg integration. The function E m is an elliptic integral, some values of which are displayed in the table : m 0 0.
Trondheim Univ. Uppsala Univ. N 18 Q 0 1 4. The result correct to five decimals is 0. The given integral becomes 0. Using Romberg integration, we obtain h O h2 O h4 O h6 method method method 0.
Methods of the form 4. The nodes and the weights for the integration method 4. We divide the interval [a, b] or [— 1, 1] into a number of subintervals and evaluate the integral in each subinterval by a particular method. The errors in the composite trapezoidal rule 4. This integral can be evaluated numerically by two successive integrations in x any y 4. Trapezoidal rule If we evaluate the inner integral in 4. Using the trapezoidal rule again in 4.
The computational molecule of the method 4. Find the error term. Stockholm Univ. Give the error estimate the values in the table are correctly rounded. Extrapolation Table h O h2 O h4 O h6 0.
Royal Inst. Newton-Cotes Methods 4. How many decimals would be required in function values? JK dx 0. Improve the results using Romberg integration. The function E m is an elliptic integral, some values of which are displayed in the table : m 0 0. Trondheim Univ. Uppsala Univ. N 18 Q 0 1 4. The result correct to five decimals is 0. The given integral becomes 0.
Using Romberg integration, we obtain h O h2 O h4 O h6 method method method 0. Give the maximal step size h to get the truncation error bound 0. Extrapolate to get a better value. Bergen Univ. Find the constant p and the error term.
State the result of using Richardson extrapolation on these values. Lund Univ. If f xi are in error atmost by 0. Gothenburg Univ. Determine A—1, A0, A1 and x1, so that the error R will be of highest possible order.
Also investigate if the order of the error is influenced by differ- ent values of the parameter a. The error term is independent of a. Answer with 4 significant digits. Oslo Univ. State In. Assume that the given integral exists. The open type formulas or a combination of open and closed type formulas discussed in the text converge very slowly.
We shall first construct quadrature rules for evaluating this integral. Hence, the result correct to 3 decimals is 1. Hence, the required value of the integral is — 0.
H x K Umea Univ. Using the methods 4. Hence, the result correct to two decimals is 0. Calculate the value of the integral correct to 3 decimal places.
Hence, the result correct to 3 decimal places is 6. Compare the results obtained with the exact solution. Using extrapolation, we obtain the better approximation as 4 3. In this case also, the method is exact.
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