f08bf computes the

Q R

factorization, with column pivoting, of a real

m

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: $m \geq 0$ .

n: Type: System..::..Int32
On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $n \geq 0$ .

a: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq \max (1, m)$
Note: the second dimension of the array a must be at least $\max (1, n)$ .
On entry: the $m$ by $n$ matrix $A$ .
On exit: if $m \geq n$ , 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 $\max (1, n)$ .
On entry: if $jpvt [j - 1] \neq 0$ , then the $j$ th column of $A$ is moved to the beginning of $A P$ 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 $k$ th column of $A$ is moved to become the $j$ th column of $A P$ ; in other words, the columns of $A P$ are the columns of $A$ in the order $jpvt [0], jpvt [1], \dots, jpvt [n - 1]$ .

tau: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array tau must be at least $\max (1, \min (m, n))$ .
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

Q R

factorization, with column pivoting, of an arbitrary rectangular real

m

n

matrix.

m \geq n

, the factorization is given by:

A P = Q (\begin{array}{r} R \\ 0 \end{array}),

where

R

is an

n

n

upper triangular matrix,

Q

is an

m

m

orthogonal matrix and

P

is an

n

n

permutation matrix. It is sometimes more convenient to write the factorization as

A P = (\begin{array}{r} Q_{1} & Q_{2} \end{array}) (\begin{array}{r} R \\ 0 \end{array}),

which reduces to

A P = Q_{1} R,

where

Q_{1}

consists of the first

n

columns of

Q

, and

Q_{2}

the remaining

m - n

columns.

m < n

R

is trapezoidal, and the factorization can be written

A P = Q (\begin{array}{r} R_{1} & R_{2} \end{array}),

where

R_{1}

is upper triangular and

R_{2}

is rectangular.

The matrix

Q

is not formed explicitly but is represented as a product of

\min (m, 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

Q R

factorization of the first

k

columns of the permuted matrix

A P

The method allows specified columns of

A

to be moved to the leading columns of

A P

at the start of the factorization and fixed there. The remaining columns are free to be interchanged so that at the

i

th stage the pivot column is chosen to be the column which maximizes the

2

-norm of elements

i

m

over columns

i

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

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision.

Parallelism and Performance

None.

Further Comments

The total number of floating-point operations is approximately

\frac{2}{3} n^{2} (3 m - n)

m \geq n

\frac{2}{3} m^{2} (3 n - m)

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

m \geq n

, 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 = Q^{T} C

, where

C

m

p

To compute a

Q R

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

\min_{x} {‖b_{j} - A x_{j}‖}_{2}, j = 1, 2

for the basic solutions

x_{1}

and

x_{2}

, where

A = (\begin{array}{r} - 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 \end{array}) and B = (\begin{array}{r} 7.4 & 2.7 \\ 4.2 & - 3.0 \\ - 8.3 & - 9.6 \\ 1.8 & 1.1 \\ 8.6 & 4.0 \\ 2.1 & - 5.7 \end{array})

and

b_{j}

is the

j

th column of the matrix

B

. The solution is obtained by first obtaining a

Q R

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

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also