Gram Schmidt Orthonormalisation - matlab

Question

I am writing the following code for Gram Schmidt Orthogonalization. It says that there's an error in calling the function. What's the error and how to rectify it?
A =[1,1,1,1;-1,4,4,-1;4,-2,2,0];
A =A';
B=myGramschmidt(A);
function [B] = myGramschmidt(A)
x1=A(:,1);
x2=A(:,2);
x3=A(:,3);
v1=x1;
c = dot(v1);
v2 = x2-((dot(x2,v1)/c)* v1);
d = dot(v2);
v3 = x3-((dot(x3,v1)/c)* v1)-((dot(x3,v2)/d)* v2);
C=[v1,v2,v3];
V1=normc(v1);
V2=normc(v2);
V3=normc(v3);
B=[V1,V2,V3];
end

Answer 1

Using the Wikipedia Gram-Schmidt page, but Luis Mendo is correct as to why you got the error.
function [B] = myGramschmidt(A)
B = A;
for k = 1:size(A, 1)
for j = 1:k-1
B(k, :) = B(k, :) - proj(B(j, :), A(k, :));
end
end
end
function p = proj(u, v)
% https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process#The_Gram.E2.80.93Schmidt_process
p = dot(v, u) / dot(u, u) * u;
end

Answer 2

Try this vectorized implementation in python.
Also I would suggest to go through David C lay book for theory.
def replace_zero(array):
for i in range(len(array)) :
if array[i] == 0 :
array[i] = 1
return array
def gram_schmidt(self,A, norm=True, row_vect=False):
"""Orthonormalizes vectors by gram-schmidt process
Parameters
-----------
A : ndarray,
Matrix having vectors in its columns
norm : bool,
Do you need Normalized vectors?
row_vect: bool,
Does Matrix A has vectors in its rows?
Returns
-------
G : ndarray,
Matrix of orthogonal vectors
Gram-Schmidt Process
--------------------
The Gram–Schmidt process is a simple algorithm for
producing an orthogonal or orthonormal basis for any
nonzero subspace of Rn.
Given a basis {x1,....,xp} for a nonzero subspace W of Rn,
define
v1 = x1
v2 = x2 - (x2.v1/v1.v1) * v1
v3 = x3 - (x3.v1/v1.v1) * v1 - (x3.v2/v2.v2) * v2
.
.
.
vp = xp - (xp.v1/v1.v1) * v1 - (xp.v2/v2.v2) * v2 - .......
.... - (xp.v(p-1) / v(p-1).v(p-1) ) * v(p-1)
Then {v1,.....,vp} is an orthogonal basis for W .
In addition,
Span {v1,.....,vp} = Span {x1,.....,xp} for 1 <= k <= p
References
----------
Linear Algebra and Its Applications - By David.C.Lay
"""
if row_vect :
# if true, transpose it to make column vector matrix
A = A.T
no_of_vectors = A.shape[1]
G = A[:,0:1].copy() # copy the first vector in matrix
# 0:1 is done to to be consistent with dimensions - [[1,2,3]]
# iterate from 2nd vector to number of vectors
for i in range(1,no_of_vectors):
# calculates weights(coefficents) for every vector in G
numerator = A[:,i].dot(G)
denominator = np.diag(np.dot(G.T,G)) #to get elements in diagonal
weights = np.squeeze(numerator/denominator)
# projected vector onto subspace G
projected_vector = np.sum(weights * G,
axis=1,
keepdims=True)
# orthogonal vector to subspace G
orthogonalized_vector = A[:,i:i+1] - projected_vector
# now add the orthogonal vector to our set
G = np.hstack((G,orthogonalized_vector))
if norm :
# to get orthoNormal vectors (unit orthogonal vectors)
# replace zero to 1 to deal with division by 0 if matrix has 0 vector
# or normazalization value comes out to be zero
G = G/self.replace_zero(np.linalg.norm(G,axis=0))
if row_vect:
return G.T
return G
G = np.array([[1,0,0],[1,1,0],[1,1,1],[1,1,1]])
gram_schmidt(G)
>
array([[ 0.5 , -0.8660254 , 0. ],
[ 0.5 , 0.28867513, -0.81649658],
[ 0.5 , 0.28867513, 0.40824829],
[ 0.5 , 0.28867513, 0.40824829]])