XploRe Help : qr

Keywords - Function groups - @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Group:	Linear Algebra
See also:	qrmat

Function:		qr
Description:		computes QR decomposition of a n x p matrix. Column pivoting based on the l_2-norms of the reduced columns may be performed at the users option.

Usage:	`{y,aux,jpvt} = qr(x,{pvt})`
Input:
	`x`	n x p matrix
	`pvt`	p x 1 vector. The vector of integers that control the selection of the pivot columns. The k-th column of x is placed in one of three classes according to the value of jpvt[k]: if pvt[k]> 0, then k-th column is an initial column, if pvt[k]= 0, then k-th column is a free column, if pvt[k] < 0, then k-th column is a final column. Before the decomposition is computed, initial columns are moved to the beginning of the matrix x and final columns to the end. Both initial and final columns are frozen in place during the computation and only free columns are moved. At the k-th stage of the reduction, if k-th column is occupied by a free column it is interchanged with the free column of largest reduced l_2-norm.
Output:
	`y`	n x p matrix. y contains in its upper triangle the upper triangular matrix R of the QR factorization. Below its diagonal x contains information from which the orthogonal part of the decomposition can be recovered.
	`aux`	p x 1 vector. aux contains further information required to recover the orthogonal part of the decomposition.
	`jpvt`	n x 1 vector. jpvt contains the index of the column of y that has been interchanged into the k-th column of the original matrix, if pivoting was requested.

Note:

of the original matrix y but that of y

with its columns permuted as described by jpvt

Example:

randomize(123) x=normal(3,4) pvt = 0.0*(1:4) {y,aux,jpvt}=qr(x,pvt) y aux jpvt

Result:

Contents of y

[1,]  -2.1492  0.34232   0.5023  -1.5094

[2,]  0.88459  -1.9599 -0.22529  0.030903

[3,] -0.40188  0.96651  -1.4159  0.43305

Contents of aux

[1,]   1.2366

[2,]   1.2566

[3,]        0

[4,]  0.43305

Contents of jpvt

[1,]        4

[2,]        2

[3,]        1

[4,]        3