sqrt(1+i)

[Albert Chan]

(10-05-2016 12:16 PM)roadrunner Wrote:  

Code:


 y=√(-ra+√(ra^2+ia^2))/√2;
 x=ia/(2*y);
 if n==1 then return x+y*i; end;
 return rootroot(x+y*i,n-1);
 ...

It crashes the emulator for values of n 6 or greater. Example:

rootroot(1+i,5) works but rootroot(1+i,6) crashes.

However, approx(rootroot(1+i,6)) returns the correct approximate answer, which tells me the program is returning a correct answer but the display can't handle it, and sits there with an hour glass in the corner.

What am I doing wrong?

Code is wrong. Think polar form (cis(θ) = cos(θ) + i*sin(θ))

z = |z| * cis(arg(z))
√z = √|z| * cis(arg(z)/2)

With arg(z) = [-pi,pi], sign(arg(z)) = sign(sin(arg(z)) → sign(im(z)) = sign(im(√z))

CAS> csqrt(z) := √((abs(z)+re(z))/2) + sign(im(z))*i*√((abs(z)-re(z))/2)
CAS> csqrt(1+i)

\(\displaystyle \sqrt{\frac{1}{2} \cdot (\sqrt{2}+1)}+ i \sqrt{\frac{1}{2} \cdot (\sqrt{2}-1)}\)

CAS> csqrt(Ans)

\(\displaystyle i \sqrt{\frac{1}{2} \cdot (-\sqrt{\frac{1}{2} \cdot (\sqrt{2}+1)}+2^{\frac{1}{4}})}+\sqrt{\frac{1}{2} \cdot (\sqrt{\frac{1}{2} \cdot (\sqrt{2}+1)}+2^{\frac{1}{4}})}\)

CAS> approx(Ans)       → 1.06955393236+0.212747504727*i
CAS> Ans^4               → 1+i

Also, with radical not simplified, mess exploded with nested square root.

CAS> rootroot(1+i, 1)

\(\displaystyle \frac{i \sqrt{\sqrt{2}-1}}{\sqrt{2}}+\frac{\sqrt{2} \cdot \frac{1}{2}}{\sqrt{\sqrt{2}-1}}\)

CAS> rootroot(Ans, 1)
\(\displaystyle \frac{1}{2} \cdot \frac{\sqrt{\sqrt{2}-1}}{\sqrt{\frac{1}{2} \cdot (2 \sqrt{\frac{1}{4} \cdot (\left(\frac{\sqrt{2}}{\sqrt{\sqrt{2}-1}}\right)^{2}+4 \left(\frac{\sqrt{\sqrt{2}-1}}{\sqrt{2}}\right)^{2})}-\frac{\sqrt{2}}{\sqrt{\sqrt{2}-1}})}}+\frac{i \sqrt{\frac{1}{2} \cdot (2 \sqrt{\frac{1}{4} \cdot (\left(\frac{\sqrt{2}}{\sqrt{\sqrt{2}-1}}\right)^{2}+4 \left(\frac{\sqrt{\sqrt{2}-1}}{\sqrt{2}}\right)^{2})}-\frac{\sqrt{2}}{\sqrt{\sqrt{2}-1}})}}{\sqrt{2}}\)

CAS> approx(Ans)       → 1.06955393236+0.212747504727*i