By Novaga M., Valdinoci E.

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Additional info for Multibump solutions and asymptotic expansions for mesoscopic Allen-Cahn type equations

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What values would be computed for x, y, and z if this code is used? 0 + 1/n output x, y, z end for c. What values would the following assignment statements produce? 1 Preliminary Remarks 19 c ← (5/9)( f − 32) f ← 9/5c + 32 output x, i, half, j, c, f d. 1416)radius output area, circum 22. Criticize the following pseudocode for evaluating limx→0 arctan(|x| )/x. Code and run it to see what happens. 0 y ← arctan(|x| )/x output x, y end for 23. Carry out some computer experiments to illustrate or test the programming suggestions in Appendix A.

Division of the semicircular arc into 2 n Next let Sn = sin θn and Pn = 2 Sn+1 . Show that Sn+1 = Sn /(2 + 2 1 − Sn ) and Pn is an approximation to π . Starting with S2 = 1 and P1 = 2, compute Sn+1 and Pn recursively for 2 n 20. 12. The irrational number π can be computed by approximating the area of a unit circle as the limit of a sequence p1 , p2 , . . described as follows. Divide the unit circle into 2n sectors. ) Approximate the area of the sector by the 38 Chapter 1 Introduction area of the isosceles triangle.

Notice that the ﬁrst application of Horner’s algorithm does not yield q in the form shown but rather as a sum of powers of x. ) This process is repeated until all coefﬁcients ck are found. We call the algorithm just described the complete Horner’s algorithm. The pseudocode for executing it is arranged so that the coefﬁcients ck overwrite the input coefﬁcients ak . q(x) = integer n, k, j; real r ; real array (ai )0:n for k = 0 to n − 1 do for j = n − 1 to k do a j ← a j + ra j+1 end for end for This procedure can be used in carrying out Newton’s method for ﬁnding roots of a polynomial, which we discuss in Chapter 3.