GEQP3
QR Factorization of a Rectangular Matrix
GEQP3
is a FORTRAN90 library which
contains the portion of the LAPACK library that carries out the
QR factorization, with column pivoting, of an M by N rectangular matrix,
with N <= M.
The factorization can be written as
A = Q * R0 * P'
= Q * (R) * P',
(0)
where
-
Q is an M by M orthogonal matrix;
-
R0 is an M by N upper trapezoidal matrix, which
is guaranteed to include a final M-N by N block of zeros.
-
R is an N by N upper triangular matrix, whose diagonal elements
are in descending magnitude, and some of which may be zero if the
system does not have full column rank N.
-
P is an N by N permutation matrix that reflects the column pivoting;
The rank of A can be estimated by the rank of R, which, in turn,
can be estimated by taking a tolerance T, and comparing the consecutive
diagonal elements of R to T * R(1,1). If K is the last diagonal element
which is at least T * R(1,1) in magnitude, we estimate the rank as K.
Thereafter, a least squares soluation of the linear system A*X=B can be
determined by:
X = P * ( inv ( R(1:k,1:k) ) * ( Q' * B )(1:k,1:k) )
( 0 )
Licensing:
The computer code and data files described and made available on this web page
are distributed under
the GNU LGPL license.
Languages:
GEQP3 is available in
a FORTRAN90 version.
Related Programs:
BAND_QR,
a FORTRAN90 library which
computes the QR factorization of a banded matrix, and can solve
related linear systems,
by Alfredo Remon, Enrique Quintana-Orti, Gregorio Quintana-Orti.
LAPACK_EXAMPLES,
a FORTRAN90 program which
demonstrates the use of the LAPACK linear algebra library.
LINPACK,
a FORTRAN90 library which
solves linear systems for a variety of matrix storage schemes,
real or complex arithmetic, and single or double precision.
It includes a routine for computing the singular value decomposition (SVD)
of a rectangular matrix.
The original version of this library is
by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.
QR_SOLVE,
a FORTRAN90 library which
computes the least squares solution of a linear system A*x=b.
References:
-
Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford,
James Demmel, Jack Dongarra, Jeremy DuCroz, Anne Greenbaum,
Sven Hammarling, Alan McKenney, Danny Sorensen,
LAPACK User's Guide,
Third Edition,
SIAM, 1999,
ISBN: 0898714478,
LC: QA76.73.F25L36
Source Code:
Examples and Tests:
List of Routines:
-
DCOPY copies a vector X to a vector Y.
-
DGEMM computes C = alpha * A * B and related operations.
-
DGEMV computes y := alpha * A * x + beta * y for general matrix A.
-
DGEQP3 computes a QR factorization with column pivoting using Level 3 BLAS.
-
DGEQRF computes a QR factorization of a real M by N matrix A = Q * R.
-
DGEQR2 computes a QR factorization of a real M by N matrix A = Q * R.
-
DGER computes A := alpha*x*y' + A.
-
DLAMC3 forces A and B to be stored prior to doing the addition.
-
DLAMCH determines real ( kind = 8 ) machine parameters.
-
DLAPY2 returns sqrt ( X^2 + Y^2 ), while avoiding unnecessary overflow.
-
DLAQP2 QR factors with column pivoting the block A(OFFSET+1:M,1:N).
-
DLAQPS computes a step of QR factorization with column pivoting.
-
DLARF applies a real elementary reflector to an M by N matrix.
-
DLARFB applies a block reflector to a general rectangular matrix.
-
DLARFG generates a real elementary reflector H of order N.
-
DLARFT forms the triangular factor T of a real block reflector H of order N.
-
DNRM2 returns the euclidean norm of a vector.
-
DORM2R overwrites M by N matrix C with Q*C, Q'*C, C*Q or C*Q'.
-
DORMQR replaces the rectangular matrix C by Q*C, Q'*C, C*Q or C*Q'.
-
DSCAL scales a vector by a constant.
-
DSWAP interchanges two vectors.
-
DTRMM performs B:=A*B or B:=B*A, A triangular, B rectangular.
-
DTRMV computes x: = A*x or x = A'*x for a triangular matrix A.
-
DTRSM performs B:=inv(A)*C or B:=C*inv(A), B and C rectangular, A triangular.
-
IDAMAX indexes the array element of maximum absolute value.
-
IEEECK verifies that infinity and NAN arithmetic are safe.
-
ILADLC scans matrix A for its last nonzero column.
-
ILADLR returns the index of the last nonzero row of a matrix.
-
ILAENV returns problem-dependent parameters.
-
IPARMQ sets problem and machine dependent parameters.
-
LSAME returns TRUE if CA is the same letter as CB regardless of case.
-
XERBLA is an error handler for the LAPACK routines.
You can go up one level to
the FORTRAN90 source codes.
Last revised on 24 March 2014.