Generate a random orthonormal vector (i) to a given unit vector and (ii) with n-2 predetermined components

Hi,

I need to create a random orthonormal vector (i) to a given unit vector and (ii) with (maximal) n-2 predetermined components. For example, say I have the unit vector v1=[-0,52 0,72 -0,19 0,37 -0,18]' (rounded to two digits) and I have the vector v2 with lets say 2 predetermined components that are non-zero, v2=[a b c 0.45 -0.09]. Thus, there should be infinite many solutions for a, b, and c such that v1`*v2=0.

I would like to find an efficient way to generate such a random orthonormal vector v2 where the elements a, b, and c are drawn from a uniform distribution.

I have tried the following:

% start with a random matrix
X = randn(3);
% add some predetermined values
predet=[.4,.5,.4;-.2,-.1,-.2];
V=[X;predet];
% normalize vectors
v1=V(:,1)/norm(V(:,1));
v2=V(:,2)/norm(V(:,2));

% find a random solution for v1'*v2==0 with 3 random entries of v2
check=0;
while check<1
    random = -1 + (1-(-1)).*rand(3,1);
    v2=[random; v2(4:5)];
    % check whether v2 is orthogonal to v1 and v2 has norm 1
    if norm(v2)>0.9999 && norm(v2)<1.0001...
            && v1'*v2<=0.0001 && v1'*v2>=-0.0001
        check=1;
    end
end

However, so far the solution is only a very bad approximation and the algorithm is very slow. Any help on this or suggestions would be very much appreciated.

[Torsten]

Just an idea:

Let v1=[v11,v12,...,v1 n], v2=[p1,p2,p3,v24,...,v2n] with unknown p1,p2,p3 satisfying

p1*v11+p2*v12+p3*v13 = -(v14*v24+...+v1n*v2n)

p1^2+p2^2+p3^2=1-(v24^2+...+v2n^2)

Thus the space to sample from is the intersection of the unit sphere and an intersecting planein 3d space.

This intersection is an ellipse.

If you are able to find a parametrization by arclength of this intersection by a parameter, you are done.

Answer 1

[darova]

What about fsolve

b = 1;
c = 1;
v1 = [-0.52 0.72 -0.19 0.37 -0.18];
v2 = @(x)[x b c 0.45 -0.09];
F = @(x) dot(v1,v2(x));
res = fsolve(F,0.5);

[Max Breitenlechner]

Hi darova,

sorry for my late response and thank you for your suggestion. Unfortunately, your solution does not work for me. The norm of v2 is not necessarily 1 and x, b, and c are not uniforly distributed if I repeat this procedure, say for N times.

I have now tried the following:

% start with a random matrix
X = randn(3);
% add some predetermined values
predet=[.4,.5,.4;-.2,-.1,-.2];
V=[X;predet];
% normalize vectors
v1=V(:,1)/norm(V(:,1));
v2=V(:,2)/norm(V(:,2));

syms a b c
v2new = [a; b; c; v2(4:5)];
assume(a > -1 & a < 1 & a~=0);
assumeAlso(b > -1 & b < 1 & b~=0);
assumeAlso(c > -1 & c < 1 & c~=0);
assumeAlso(norm(v2new) ==1);
assumptions
solution=solve(dot(v1,v2new)==0,[a b c],'ReturnConditions',true); 

But here I get the following error message: "Error using mupadengine/feval (line 163) The row index is out of range."

But there should exist (infinite many) solutions as the above described algorithm finds a solution for every draw.

Any ideas why I get this error message or how I can obtain a random solution under multiple constraints?

Thanks!

[darova]

Try this

v1 = [-0.52 0.72 -0.19 0.37 -0.18];
v2 = @(x,b,c)[x b c 0.45 -0.09];
F = @(x,b,c) dot(v1,v2(x,b,c));

bb = -10:10;
cc = -10:10;
[B,C] = meshgrid(bb,cc);
A = B*0;
res = 0.5;
opt = optimset('display','off');
for i = 1:size(A,1)
    for j = 1:size(A,2)
        res = fsolve(F,res,opt, B(i,j),C(i,j));
        A(i,j) = res;
    end
end
surf(B,C,A)
xlabel('B')
ylabel('C')
zlabel('A')

[Max Breitenlechner]

Thank you - that's an interesting solution!