Let a number z* be an approximation of a number r and the corresponding relative error is equal to n > 0. One of the standard, and widely used, ways to measure accuracy of the approximation r = r* is to find the corresponding number of significant digits. By the definition, the approximation 12 r* has k significant digits if k is the maximal natural number such that RE(r = r*)=n 5 104 4. (Floating-Point Arithmetic; Taylor Polynomials). It is known that * = 4.arctan(33/74) - arctan(-41/107)). Next, the Taylor (Maclaurin) polynomial of degree nine for the arctangent function about zero is 23 27 T(x) = x- 5 7 + 3 (i) Find an approximation y of T 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 (FPA7). 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 55 -17:7 + X99:9 + 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 1 to find the relative error and the number of significant digits in the approximation yy of y by y*. Please give a complete solution to the problem. Let a number z* be an approximation of a number r and the corresponding relative error is equal to n > 0. One of the standard, and widely used, ways to measure accuracy of the approximation r = r* is to find the corresponding number of significant digits. By the definition, the approximation 12 r* has k significant digits if k is the maximal natural number such that RE(r = r*)=n 5 104 4. (Floating-Point Arithmetic; Taylor Polynomials). It is known that * = 4.arctan(33/74) - arctan(-41/107)). Next, the Taylor (Maclaurin) polynomial of degree nine for the arctangent function about zero is 23 27 T(x) = x- 5 7 + 3 (i) Find an approximation y of T 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 (FPA7). 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 55 -17:7 + X99:9 + 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 1 to find the relative error and the number of significant digits in the approximation yy of y by y*. Please give a complete solution to the
