Gram–Schmidt process

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(Redirected from Gram-Schmidt orthogonalization)

In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rⁿ. The Gram–Schmidt process takes a finite, linearly independent set S = {v₁, …, v_k} for k ≤ n and generates an orthogonal set S' = {u₁, …, u_k} that spans the same k-dimensional subspace of Rⁿ as S.

The method is named for Jørgen Pedersen Gram and Erhard Schmidt but it appeared earlier in the work of Laplace and Cauchy. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition.

The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).

[edit] The Gram–Schmidt process

We define the projection operator by

$\mathrm{proj}_{\mathbf{u}}\,(\mathbf{v}) = {\langle \mathbf{u}, \mathbf{v}\rangle\over\langle \mathbf{u}, \mathbf{u}\rangle}\mathbf{u} = {\langle \mathbf{u}, \mathbf{v}\rangle} {\mathbf{u}\over\langle \mathbf{u}, \mathbf{u}\rangle},$

where <u, v> denotes the inner product of the vectors u and v. This operator projects the vector v orthogonally onto the vector u.

The Gram–Schmidt process then works as follows:

	$\mathbf{u}_1 = \mathbf{v}_1,$		$\mathbf{e}_1 = {\mathbf{u}_1 \over \\|\mathbf{u}_1\\|}$
	$\mathbf{u}_2 = \mathbf{v}_2-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_2),$		$\mathbf{e}_2 = {\mathbf{u}_2 \over \\|\mathbf{u}_2\\|}$
	$\mathbf{u}_3 = \mathbf{v}_3-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_3)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_3),$		$\mathbf{e}_3 = {\mathbf{u}_3 \over \\|\mathbf{u}_3\\|}$
	$\mathbf{u}_4 = \mathbf{v}_4-\mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_4)-\mathrm{proj}_{\mathbf{u}_3}\,(\mathbf{v}_4),$		$\mathbf{e}_4 = {\mathbf{u}_4 \over \\|\mathbf{u}_4\\|}$
	$\vdots$		$\vdots$
	$\mathbf{u}_k = \mathbf{v}_k-\sum_{j=1}^{k-1}\mathrm{proj}_{\mathbf{u}_j}\,(\mathbf{v}_k),$		$\mathbf{e}_k = {\mathbf{u}_k\over \\|\mathbf{u}_k \\|}$

The first two steps of the Gram–Schmidt process.

The sequence u₁, …, u_k is the required system of orthogonal vectors, and the normalized vectors e₁, …, e_k form an orthonormal set. The calculation of the sequence u₁, …, u_k is known as Gram–Schmidt orthogonalization, whilst the calculation of the sequence e₁, …, e_k is known as Gram–Schmidt orthonormalization as the vectors are normalized.

To check that these formulas yield an orthogonal sequence, first compute 〈u₁, u₂〉 by substituting the above formula for u₂: we get zero. Then use this to compute 〈u₁, u₃〉 again by substituting the formula for u₃: we get zero. The general proof proceeds by mathematical induction.

Geometrically, this method proceeds as follows: to compute u_i, it projects v_i orthogonally onto the subspace U generated by u₁, …, u_i−1, which is the same as the subspace generated by v₁, …, v_i−1. The vector u_i is then defined to be the difference between v_i and this projection, guaranteed to be orthogonal to all of the vectors in the subspace U.

The Gram–Schmidt process also applies to a linearly independent infinite sequence {v_i}_i. The result is an orthogonal (or orthonormal) sequence {u_i}_i such that for natural number n: the algebraic span of v₁, …, v_n is the same as that of u₁, …, u_n.

If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on the $i$ th step, assuming that $\mathbf{v_i}$ is a linear combination of $\mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{i-1}}$ . If this can happen, then in order to meet the condition that the outputs be orthogonal, and furthermore to avoid division by zero if producing an orthonormal basis, the algorithm should test for zero vectors in the output and discard them. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs.

[edit] Example

Consider the following set of vectors in R² (with the conventional inner product)

$S = \left\lbrace\mathbf{v}_1=\begin{pmatrix} 3 \\ 1\end{pmatrix}, \mathbf{v}_2=\begin{pmatrix}2 \\2\end{pmatrix}\right\rbrace.$

Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors:

$\mathbf{u}_1=\mathbf{v}_1=\begin{pmatrix}3\\1\end{pmatrix}$

$\mathbf{u}_2 = \mathbf{v}_2 - \mathrm{proj}_{\mathbf{u}_1} \, (\mathbf{v}_2) = \begin{pmatrix}2\\2\end{pmatrix} - \mathrm{proj}_{({3 \atop 1})} \, ({\begin{pmatrix}2\\2\end{pmatrix})} = \begin{pmatrix} -2/5 \\6/5 \end{pmatrix}.$

We check that the vectors u₁ and u₂ are indeed orthogonal:

$\langle\mathbf{u}_1,\mathbf{u}_2\rangle = \left\langle \begin{pmatrix}3\\1\end{pmatrix}, \begin{pmatrix}-2/5\\6/5\end{pmatrix} \right\rangle = -\frac65 + \frac65 = 0.$

We can then normalize the vectors by dividing out their sizes as shown above:

$\mathbf{e}_1 = {1 \over \sqrt {10}}\begin{pmatrix}3\\1\end{pmatrix}$

$\mathbf{e}_2 = {1 \over \sqrt{40 \over 25}} \begin{pmatrix}-2/5\\6/5\end{pmatrix} = {1\over\sqrt{10}} \begin{pmatrix}-1\\3\end{pmatrix}.$

[edit] Numerical stability

When this process is implemented on a computer, the vectors $u k$ are often not quite orthogonal, due to rounding errors. For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt") this loss of orthogonality is particularly bad; therefore, it is said that the (classical) Gram–Schmidt process is numerically unstable.

The Gram–Schmidt process can be stabilized by a small modification. Instead of computing the vector u_k as

$\mathbf{u}_k = \mathbf{v}_k - \mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_k) - \mathrm{proj}_{\mathbf{u}_2}\,(\mathbf{v}_k) - \cdots - \mathrm{proj}_{\mathbf{u}_{k-1}}\,(\mathbf{v}_k),$

it is computed as

$\begin{align} \mathbf{u}_k^{(1)} &= \mathbf{v}_k - \mathrm{proj}_{\mathbf{u}_1}\,(\mathbf{v}_k), \\ \mathbf{u}_k^{(2)} &= \mathbf{u}_k^{(1)} - \mathrm{proj}_{\mathbf{u}_2} \, (\mathbf{u}_k^{(1)}), \\ & \,\,\, \vdots \\ \mathbf{u}_k^{(k-2)} &= \mathbf{u}_k^{(k-3)} - \mathrm{proj}_{\mathbf{u}_{k-2}} \, (\mathbf{u}_k^{(k-3)}), \\ \mathbf{u}_k &= \mathbf{u}_k^{(k-2)} - \mathrm{proj}_{\mathbf{u}_{k-1}} \, (\mathbf{u}_k^{(k-2)}). \end{align}$

This approach (sometimes referred to as "modified Gram–Schmidt") gives the same result as the original formula in exact arithmetic, but it introduces smaller errors in finite-precision arithmetic.

[edit] Algorithm

The following algorithm implements the stabilized Gram–Schmidt orthonormalization. The vectors v₁, …, v_k are replaced by orthonormal vectors which span the same subspace.

for j from 1 to k do

for i from 1 to j − 1 do

$\mathbf{v}_j \leftarrow \mathbf{v}_j - \mathrm{proj}_{\mathbf{v}_{i}} \, \mathbf{v}_j$ (remove component in direction v_i)

next i

$\mathbf{v}_j \leftarrow \frac{\mathbf{v}_j}{\|\mathbf{v}_j\|}$ (normalize)

next j

The cost of this algorithm is asymptotically 2nk² floating point operations, where n is the dimensionality of the vectors (Golub & Van Loan 1996, §5.2.8).

[edit] Alternatives

Other orthogonalization algorithms use Householder transformations or Givens rotations. The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the $j$ th orthogonalized vector after the $j$ th iteration, while orthogonalization using Householder reflections produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for iterative methods like the Arnoldi iteration.

[edit] References

Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9 .
Bau III, David; Trefethen, Lloyd N. (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-361-9 .

[edit] External links

Mathematics portal

Harvey Mudd College Math Tutorial on the Gram-Schmidt algorithm
MIT Linear Algebra Lecture on Gram-Schmidt
Earliest known uses of some of the words of mathematics: G The entry "Gram-Schmidt orthogonalization" has some information and references on the origins of the method.
Demos: Gram Schmidt process in plane and Gram Schmidt process in space

Gram–Schmidt process

From Wikipedia, the free encyclopedia

Contents

[edit] The Gram–Schmidt process

[edit] Example

[edit] Numerical stability

[edit] Algorithm

[edit] Alternatives

[edit] References

[edit] External links

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