naginterfaces.library.orthog.real_​gram_​schmidt

naginterfaces.library.orthog.real_gram_schmidt(a, n1)

real_gram_schmidt applies the Schmidt orthogonalization process to $n$ vectors in $m$-dimensional space, $n\le m$.

For full information please refer to the NAG Library document for f05aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f05/f05aaf.html

Parameters

afloat, array-like, shape $(m,\text{n2})$

Columns $n1$ to $\text{n2}$ contain the vectors to be orthogonalized. The vectors are stored by columns in elements $1$ to $m$.

n1int

The indices of the first and last columns of $A$ to be orthogonalized.

Returns

afloat, ndarray, shape $(m,\text{n2})$

These vectors are overwritten by the orthonormal vectors.

ccfloat

Is used to indicate linear dependence of the original vectors. The nearer $cc$ is to $1.0$, the more likely vector $icol$ is dependent on vectors $n1$ to $icol-1$. See Further Comments.

icolint

The column number corresponding to $cc$. See Further Comments.

Raises

NagValueError

(errno $1$)

On entry, $n1=⟨value⟩$ and $\text{n2}=⟨value⟩$.

Constraint: $n1\le \text{n2}$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

real_gram_schmidt applies the Schmidt orthogonalization process to $n$ linearly independent vectors in $m$-dimensional space, $n\le m$. The effect of this process is to replace the original $n$ vectors by $n$ orthonormal vectors which have the property that the $\text{r}$th vector is linearly dependent on the first $\text{r}$ of the original vectors, and that the sum of squares of the elements of the $\text{r}$th vector is equal to $1$, for $\text{r}=1,2,\dots ,n$. Inner-products are accumulated using additional precision.