NAG Library Routine Document
F08AEF (DGEQRF)
1 Purpose
F08AEF (DGEQRF) computes the QR factorization of a real m by n matrix.
2 Specification
INTEGER |
M, N, LDA, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), TAU(*), WORK(max(1,LWORK)) |
|
The routine may be called by its
LAPACK
name dgeqrf.
3 Description
F08AEF (DGEQRF) forms the QR factorization of an arbitrary rectangular real m by n matrix. No pivoting is performed.
If
m≥n, the factorization is given by:
where
R is an
n by
n upper triangular matrix and
Q is an
m by
m orthogonal matrix. It is sometimes more convenient to write the factorization as
which reduces to
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
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 Chapter Introduction for details). Routines are provided to work with
Q in this representation (see
Section 8).
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 original matrix
A.
4 References
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint:
M≥0.
- 2: N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint:
N≥0.
- 3: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
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
m≥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.
- 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08AEF (DGEQRF) is called.
Constraint:
LDA≥max1,M.
- 5: TAU(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
TAU
must be at least
max1,minM,N.
On exit: further details of the orthogonal matrix Q.
- 6: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 7: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08AEF (DGEQRF) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Suggested value:
for optimal performance, LWORK≥N×nb, where nb is the optimal block size.
Constraint:
LWORK≥max1,N or LWORK=-1.
- 8: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
7 Accuracy
The computed factorization is the exact factorization of a nearby matrix
A+E, where
and
ε is the
machine precision.
8 Further Comments
The total number of floating point operations is approximately
23
n2
3m-n
if m≥n or
23
m2
3n-m
if m<n.
To form the orthogonal matrix
Q F08AEF (DGEQRF) may be followed by a call to
F08AFF (DORGQR):
CALL DORGQR(M,M,MIN(M,N),A,LDA,TAU,WORK,LWORK,INFO)
but note that the second dimension of the array
A must be at least
M, which may be larger than was required by F08AEF (DGEQRF).
When
m≥n, it is often only the first
n columns of
Q that are required, and they may be formed by the call:
CALL DORGQR(M,N,N,A,LDA,TAU,WORK,LWORK,INFO)
To apply
Q to an arbitrary real rectangular matrix
C, F08AEF (DGEQRF) may be followed by a call to
F08AGF (DORMQR). For example,
CALL DORMQR('Left','Transpose',M,P,MIN(M,N),A,LDA,TAU,C,LDC,WORK, &
LWORK,INFO)
forms
C=QTC, where
C is
m by
p.
To compute a
QR factorization with column pivoting, use
F08BEF (DGEQPF).
The complex analogue of this routine is
F08ASF (ZGEQRF).
9 Example
This example solves the linear least squares problems
where
b1 and
b2 are the columns of the matrix
B,
9.1 Program Text
Program Text (f08aefe.f90)
9.2 Program Data
Program Data (f08aefe.d)
9.3 Program Results
Program Results (f08aefe.r)