Page 42 - Numerical methods for chemical engineering

28 1 Linear algebra

b 2
w 2

b 1 w 1
str act tis
ar t r b 2
Figure 1.4 Gram–Schmidt method makes vectors orthogonal through projection operations.

Gram–Schmidt orthogonalization
[1]
We describe here a simple method for forming an orthogonal basis set {w ,
N
[2]
[1]
[2]
w ,..., w [N] } for from one, {b , b ,..., b [N] }, that is merely linearly indepen-
dent. As a first step, we simply assign w [1] to be
w [1] = b [1] (1.141)
[1]
Next, we write w [2] as a linear combination of b [2] and w ,
w [2] = b [2] + s 21 w [1] s 21 ∈ (1.142)

and enforce that w [2] be orthogonal to w [1] through our choice of s 21 ,
w [1] · b [2]
[1]
[1]
[2]
[2]
[1]
[1]

w · w = 0 = w · b + s 21 w · w (1.143)
s 21 =−
2
w
[1]
so that

[1] [2]
w · b [1]
[2]
[2]
w = b − w (1.144)
2
w
[1]
[1]
Essentially, we “project out” the component of b [2] in the direction of w , as shown in
Figure 1.4.
We next write
[3] [3] [1] [2]
w = b + s 31 w + s 32 w (1.145)
and choose s 31 and s 32 such that w [3] · w [1] = w [3] · w [2] = 0, to obtain

w [1] · b [3] [1] w [2] · b [3] [2]
[3]
[3]
w = b − w − w (1.146)
2 2
w [1] w [2]
[1]
[2]
This process may be continued to fill out the orthogonal basis set {w , w ,..., w [N] },
where

k−1 [ j] [k]
[k] [k] w · b [ j]
w = b − w (1.147)
2
j=1 w [ j]