f08bf computes the QR factorization, with column pivoting, of a real m by n matrix.

Syntax

C#
public static void f08bf(
	int m,
	int n,
	double[,] a,
	int[] jpvt,
	double[] tau,
	out int info
)
Visual Basic
Public Shared Sub f08bf ( _
	m As Integer, _
	n As Integer, _
	a As Double(,), _
	jpvt As Integer(), _
	tau As Double(), _
	<OutAttribute> ByRef info As Integer _
)
Visual C++
public:
static void f08bf(
	int m, 
	int n, 
	array<double,2>^ a, 
	array<int>^ jpvt, 
	array<double>^ tau, 
	[OutAttribute] int% info
)
F#
static member f08bf : 
        m : int * 
        n : int * 
        a : float[,] * 
        jpvt : int[] * 
        tau : float[] * 
        info : int byref -> unit 

Parameters

m
Type: System..::..Int32
On entry: m, the number of rows of the matrix A.
Constraint: m0.
n
Type: System..::..Int32
On entry: n, the number of columns of the matrix A.
Constraint: n0.
a
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: dim1max1,m
Note: the second dimension of the array a must be at least max1,n.
On entry: the m by n matrix A.
On exit: if mn, the elements below the diagonal are overwritten by details of the orthogonal matrix Q and the upper triangle is overwritten by the corresponding elements of the n by n upper triangular matrix R.
If m<n, the strictly lower triangular part is overwritten by details of the orthogonal matrix Q and the remaining elements are overwritten by the corresponding elements of the m by n upper trapezoidal matrix R.
jpvt
Type: array<System..::..Int32>[]()[][]
An array of size [dim1]
Note: the dimension of the array jpvt must be at least max1,n.
On entry: if jpvt[j-1]0, then the j th column of A is moved to the beginning of AP before the decomposition is computed and is fixed in place during the computation. Otherwise, the j th column of A is a free column (i.e., one which may be interchanged during the computation with any other free column).
On exit: details of the permutation matrix P. More precisely, if jpvt[j-1]=k, then the kth column of A is moved to become the j th column of AP; in other words, the columns of AP are the columns of A in the order jpvt[0],jpvt[1],,jpvt[n-1].
tau
Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array tau must be at least max1,minm,n.
On exit: the scalar factors of the elementary reflectors.
info
Type: System..::..Int32%
On exit: info=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

f08bf forms the QR factorization, with column pivoting, of an arbitrary rectangular real m by n matrix.
If mn, the factorization is given by:
AP=QR0,
where R is an n by n upper triangular matrix, Q is an m by m orthogonal matrix and P is an n by n permutation matrix. It is sometimes more convenient to write the factorization as
AP=Q1Q2R0,
which reduces to
AP=Q1R,
where Q1 consists of the first n columns of Q, and Q2 the remaining m-n columns.
If m<n, R is trapezoidal, and the factorization can be written
AP=QR1R2,
where R1 is upper triangular and R2 is rectangular.
The matrix Q is not formed explicitly but is represented as a product of minm,n elementary reflectors (see the F08 class for details). Methods are provided to work with Q in this representation (see [Further Comments]).
Note also that for any k<n, the information returned in the first k columns of the array a represents a QR factorization of the first k columns of the permuted matrix AP.
The method allows specified columns of A to be moved to the leading columns of AP at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the ith stage the pivot column is chosen to be the column which maximizes the 2-norm of elements i to m over columns i to n.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Error Indicators and Warnings

Some error messages may refer to parameters that are dropped from this interface (LDA) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
ifail=-9000
An error occured, see message report.
ifail=-6000
Invalid Parameters value
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value
ifail=-6000
Invalid Parameters value

Accuracy

The computed factorization is the exact factorization of a nearby matrix A+E, where
E2=OεA2,
and ε is the machine precision.

Parallelism and Performance

None.

Further Comments

The total number of floating-point operations is approximately 23n23m-n if mn or 23m23n-m if m<n.
To form the orthogonal matrix Q f08bf may be followed by a call to (F08AFF not in this release): but note that the second dimension of the array a must be at least m, which may be larger than was required by f08bf.
When mn, it is often only the first n columns of Q that are required, and they may be formed by the call:
To apply Q to an arbitrary real rectangular matrix C, f08bf may be followed by a call to f08ag. For example, forms C=QTC, where C is m by p.
To compute a QR factorization without column pivoting, use (F08AEF not in this release).
The complex analogue of this method is (F08BTF not in this release).

Example

This example solves the linear least squares problems
minxbj-Axj2,  j=1,2
for the basic solutions x1 and x2, where
A= -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and  B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7
and bj is the jth column of the matrix B. The solution is obtained by first obtaining a QR factorization with column pivoting of the matrix A. A tolerance of 0.01 is used to estimate the rank of A from the upper triangular factor, R.

Example program (C#): f08bfe.cs

Example program data: f08bfe.d

Example program results: f08bfe.r

See Also