By Daoud Sulaiman Mashat

A numerical strategy for quasiconformal mapping of doubly hooked up domain names onto

annuli is gifted. The annulus itself isn't really identified a priori and is decided as

part of the answer strategy. The numerical approach calls for fixing a sequence

of inhomogeneous Beltrami equations, each one inside of a unique annulus, in an iterative

mode. The annulus during which the equation is being solved can be updated

during the iterations utilizing an updating strategy in response to the bisection method.

This quasiconformal mapping technique relies on Daripa's approach to quasiconformal

mapping of easily hooked up domain names onto unit disks. The functionality of

the quasiconformal mapping set of rules has been proven on a number of doubly

connected domain names with varied complicated dilations.

The answer of the Beltrami equation in an annulus calls for comparing two

singular vital operators. speedy algorithms for his or her exact evaluate are presented.

These are in response to extension of a quick set of rules of Daripa. those algorithms

are in keeping with a few recursive family members in Fourier house and the FFT (fast

Fourier transform), and feature theoretical computational complexity of order log N

per aspect.

**Read Online or Download Fast Algorithms and Their Applications to Numerical Quasiconformal Mappings of Doubly Connected Domains Onto Annuli PDF**

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**Additional resources for Fast Algorithms and Their Applications to Numerical Quasiconformal Mappings of Doubly Connected Domains Onto Annuli**

**Example text**

C_ 1 (r1) = 2, then set Bs,2(rl) = hs+2(rr) for l E [2, M) and s E Step 8. For l E (1, M) and s E [-Q, Q- m). Set Ss,m(rt) Step 9. OUTPUT Tmh(a = ne 2 rrik/N) [-Q, Q- 2). = Bs,m(rl) + Cs,m(rr). ::: Ss,m(rt)e 2rriks/N. s=-Q STOP. 1. N must be a power of two. 2. Q must be greater than 2. 2. The Algorithmic Complexity We consider the computational complexity of the above algorithm. Below We discuss the asymptotic operation count and asymptotic storage requirement. A brief analysis of the algorithmic complexity follows.

Step 4. Compute Cs,m(rz) for s E {[-Q, -m] U [0, Q -m]} and l E [1, M] as follows: Set Cs,M(rM) = 0 for all s E [0, Q- m] do l = M - 1, ... , 1 19 end do Set Cs,m(rl) = 0 for all s E [-Q, -m) dol= 2, ... ,M end do = 2, then set If m = 1, then set Step 5. If m Step 6. Step 7. If m = 0 for l E [1, M). Bs,I(rt) = 0, s E [-Q, Q- 1). C_ 1 (r1) = 2, then set Bs,2(rl) = hs+2(rr) for l E [2, M) and s E Step 8. For l E (1, M) and s E [-Q, Q- m). Set Ss,m(rt) Step 9. OUTPUT Tmh(a = ne 2 rrik/N) [-Q, Q- 2).

33 TABLE 3 Comparison of the inner radius of the annulus when N = 256 for the domain in Example 12 with ). = ). 27 X X 10- 4 10- 4 Iteration I 17 16 16 Iteration II 15 15 15 TABLE 4 Comparison of the inner radius of the annulus when M = 51 for the domain in Example 12 with ). = ). 01355 Iteration I 17 10 9 Iteration II 15 15 15 34 Figure 3. Quasiconformal mapprng of the illterior of the annulus onto the illterior of the doubly connected domarn G for Example 1. 0 Figure 4. Convergence of F(Ro) for Example 1.