Let a number * be an approximation of a number az and the corresponding relative error...
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Let a number * be an approximation of a number az and the corresponding relative error is equal to n > 0. One of the standard, and widely used, ways to measure accuracy of the approximation is to find the corresponding number of significant digits. By the definition, the approximation ≈r has k significant digits if k is the maximal natural number such that RE(za*)= n < 4. (Floating-Point Arithmetic; Taylor Polynomials). It is known that π = 4. (arctan (33/74) - arctan(-41/107)). 10% Next, the Taylor (Maclaurin) polynomial of degree nine for the arctangent function about zero is 25 27 5 T(x)=x- + (i) Find an approximation y* of as y =4-(T(33/74) - T(-41/107)), as suggested by the above reasoning. All calculations are to be carried out in the 7-digit floating-point arithmetic (FPA). To evaluate the representation T(33/74) in the FPA, the with a scientific calculator, perform the following steps: (1) switch to (Sci 7) mode, (2) type and execute the following commands one by one: 33÷74 → X X - x^3÷3 + x^5÷5 - x^7÷7+X^9÷9 → A After the above program is executed, the required value will be stored in the variable A. Work similarly, to find the representation of T(-41/107) in the FPA7, but, this time, store the resulting value in the variable B. Finish the calculations by executing the command 4 x (A B) → Y. (ii) Use the 12-digit representation y = 3.14159265359 of a to find the relative error and the number of significant digits in the approximation yy of y by y. Let a number * be an approximation of a number az and the corresponding relative error is equal to n > 0. One of the standard, and widely used, ways to measure accuracy of the approximation is to find the corresponding number of significant digits. By the definition, the approximation ≈r has k significant digits if k is the maximal natural number such that RE(za*)= n < 4. (Floating-Point Arithmetic; Taylor Polynomials). It is known that π = 4. (arctan (33/74) - arctan(-41/107)). 10% Next, the Taylor (Maclaurin) polynomial of degree nine for the arctangent function about zero is 25 27 5 T(x)=x- + (i) Find an approximation y* of as y =4-(T(33/74) - T(-41/107)), as suggested by the above reasoning. All calculations are to be carried out in the 7-digit floating-point arithmetic (FPA). To evaluate the representation T(33/74) in the FPA, the with a scientific calculator, perform the following steps: (1) switch to (Sci 7) mode, (2) type and execute the following commands one by one: 33÷74 → X X - x^3÷3 + x^5÷5 - x^7÷7+X^9÷9 → A After the above program is executed, the required value will be stored in the variable A. Work similarly, to find the representation of T(-41/107) in the FPA7, but, this time, store the resulting value in the variable B. Finish the calculations by executing the command 4 x (A B) → Y. (ii) Use the 12-digit representation y = 3.14159265359 of a to find the relative error and the number of significant digits in the approximation yy of y by y.
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Answer To find the approximation u of using the Taylor polynomial Tx we need to evaluate T3374 and T... View the full answer
Related Book For
Complex Variables and Applications
ISBN: 978-0073051949
8th edition
Authors: James Brown, Ruel Churchill
Posted Date:
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