NAG Library Routine Document
F02WDF
1 Purpose
F02WDF returns the Householder $QU$ factorization of a real rectangular $m$ by $n$ $\left(m\ge n\right)$ matrix $A$. Further, on request or if $A$ is not of full rank, part or all of the singular value decomposition of $A$ is returned.
2 Specification
SUBROUTINE F02WDF ( 
M, N, A, LDA, WANTB, B, TOL, SVD, IRANK, Z, SV, WANTR, R, LDR, WANTPT, PT, LDPT, WORK, LWORK, IFAIL) 
INTEGER 
M, N, LDA, IRANK, LDR, LDPT, LWORK, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,N), B(M), TOL, Z(N), SV(N), R(LDR,N), PT(LDPT,N), WORK(LWORK) 
LOGICAL 
WANTB, SVD, WANTR, WANTPT 

3 Description
The real
$m$ by
$n$ $\left(m\ge n\right)$ matrix
$A$ is first factorized as
where
$Q$ is an
$m$ by
$m$ orthogonal matrix and
$U$ is an
$n$ by
$n$ upper triangular matrix.
If either
$U$ is singular or
SVD is supplied as .TRUE., then the singular value decomposition (SVD) of
$U$ is obtained so that
$U$ is factorized as
where
$R$ and
$P$ are
$n$ by
$n$ orthogonal matrices and
$D$ is the
$n$ by
$n$ diagonal matrix
with
$s{v}_{1}\ge s{v}_{2}\ge \cdots \ge s{v}_{n}\ge 0\text{.}$Note that the SVD of
$A$ is then given by
the diagonal elements of
$D$ being the singular values of
$A$.
The option to form a vector ${Q}^{\mathrm{T}}b$, or if appropriate ${Q}_{1}^{\mathrm{T}}b$, is also provided.
The rank of the matrix
$A$, based upon a usersupplied parameter
TOL, is also returned.
The $QU$ factorization of $A$ is obtained by Householder transformations. To obtain the SVD of $U$ the matrix is first reduced to bidiagonal form by means of plane rotations and then the $QR$ algorithm is used to obtain the SVD of the bidiagonal form.
4 References
Wilkinson J H (1978) Singular Value Decomposition – Basic Aspects Numerical Software – Needs and Availability (ed D A H Jacobs) Academic Press
5 Parameters
 1: M – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint:
${\mathbf{M}}\ge {\mathbf{N}}$.
 2: N – INTEGERInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint:
$1\le {\mathbf{N}}\le {\mathbf{M}}$.
 3: A(LDA,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the leading
$m$ by
$n$ part of
A must contain the matrix to be factorized.
On exit: the leading
$m$ by
$n$ part of
A, together with the
$n$element vector
Z, contains details of the Householder
$QU$ factorization.
Details of the storage of the
$QU$ factorization are given in
Section 8.4.
 4: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F02WDF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{M}}$.
 5: WANTB – LOGICALInput
On entry: must be .TRUE. if
${Q}^{\mathrm{T}}b$ or
${Q}_{1}^{\mathrm{T}}b$ is required.
If on entry
${\mathbf{WANTB}}=\mathrm{.FALSE.}$,
B is not referenced.
 6: B(M) – REAL (KIND=nag_wp) arrayInput/Output
On entry: if
WANTB is supplied as .TRUE.,
B must contain the
$m$ element vector
$b$. Otherwise,
B is not referenced.
On exit: contains
${Q}_{1}^{\mathrm{T}}b$ if
SVD is returned as .TRUE. and
${Q}^{\mathrm{T}}b$ if
SVD is returned as .FALSE..
 7: TOL – REAL (KIND=nag_wp)Input
On entry: must specify a relative tolerance to be used to determine the rank of
$A$.
TOL should be chosen as approximately the largest relative error in the elements of
$A$. For example, if the elements of
$A$ are correct to about
$4$ significant figures,
TOL should be set to about
$5\times {10}^{4}$. See
Section 8.3 for a description of how
TOL is used to determine rank.
If
TOL is outside the range
$\left(\epsilon ,1.0\right)$, where
$\epsilon $ is the
machine precision, the value
$\epsilon $ is used in place of
TOL. For most problems this is unreasonably small.
 8: SVD – LOGICALInput/Output
On entry: must be .TRUE. if the singular values are to be found even if
$A$ is of full rank.
If before entry, ${\mathbf{SVD}}=\mathrm{.FALSE.}$ and $A$ is determined to be of full rank, only the $QU$ factorization of $A$ is computed.
On exit: is returned as .FALSE. if only the $QU$ factorization of $A$ has been obtained and is returned as .TRUE. if the singular values of $A$ have been obtained.
 9: IRANK – INTEGEROutput
On exit: returns the rank of the matrix
$A$. (It should be noted that it is possible for
IRANK to be returned as
$n$ and
SVD to be returned as .TRUE., even if
SVD was supplied as .FALSE.. This means that the matrix
$U$ only just failed the test for nonsingularity.)
 10: Z(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the
$n$element vector
Z contains some details of the Householder transformations. See
Section 8.4 for further information.
 11: SV(N) – REAL (KIND=nag_wp) arrayOutput
On exit: if
SVD is returned as .TRUE.,
SV contains the
$n$ singular values of
$A$ arranged in descending order.
 12: WANTR – LOGICALInput
On entry: must be .TRUE. if the orthogonal matrix
$R$ is required when the singular values are computed.
If on entry
${\mathbf{WANTR}}=\mathrm{.FALSE.}$,
R is not referenced.
 13: R(LDR,N) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array
R
must be at least
${\mathbf{N}}$ if
${\mathbf{WANTR}}=\mathrm{.TRUE.}$, and at least
$1$ otherwise.
On exit: if
SVD is returned as .TRUE. and
WANTR was supplied as .TRUE., the leading
$n$ by
$n$ part of
R will contain the lefthand orthogonal matrix of the
SVD of
$U$.
 14: LDR – INTEGERInput
On entry: the first dimension of the array
R as declared in the (sub)program from which F02WDF is called.
Constraints:
 if ${\mathbf{WANTR}}=\mathrm{.TRUE.}$, ${\mathbf{LDR}}\ge {\mathbf{N}}$;
 otherwise $1$.
 15: WANTPT – LOGICALInput
On entry: must be .TRUE. if the orthogonal matrix
${P}^{\mathrm{T}}$ is required when the singular values are computed.
Note that if
SVD is returned as .TRUE.,
PT is referenced even if
WANTPT is supplied as .FALSE., but see parameter
PT.
 16: PT(LDPT,N) – REAL (KIND=nag_wp) arrayOutput
On exit: if
SVD is returned as .TRUE. and
WANTPT was supplied as .TRUE., the leading
$n$ by
$n$ part of
PT contains the orthogonal matrix
${P}^{\mathrm{T}}$.
If
SVD is returned as .TRUE., but
WANTPT was supplied as .FALSE., the leading
$n$ by
$n$ part of
PT is used for internal workspace.
 17: LDPT – INTEGERInput
On entry: the first dimension of the array
PT as declared in the (sub)program from which F02WDF is called.
Constraint:
${\mathbf{LDPT}}\ge {\mathbf{N}}$.
 18: WORK(LWORK) – REAL (KIND=nag_wp) arrayOutput
On exit: if
SVD is returned as .FALSE.,
${\mathbf{WORK}}\left(1\right)$ contains the condition number
${\Vert U\Vert}_{E}{\Vert {U}^{1}\Vert}_{E}$ of the upper triangular matrix
$U$.
If
SVD is returned as .TRUE.,
${\mathbf{WORK}}\left(1\right)$ will contain the total number of iterations taken by the
$QR$ algorithm.
The rest of the array is used as workspace and so contains no meaningful information.
 19: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F02WDF is called.
Constraint:
${\mathbf{LWORK}}\ge 3\times {\mathbf{N}}$.
 20: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{N}}<1$, 
or  ${\mathbf{M}}<{\mathbf{N}}$, 
or  ${\mathbf{LDA}}<{\mathbf{M}}$, 
or  ${\mathbf{LDR}}<{\mathbf{N}}$ when ${\mathbf{WANTR}}=\mathrm{.TRUE.}$, 
or  ${\mathbf{LDPT}}<{\mathbf{N}}$ 
or  ${\mathbf{LWORK}}<3\times {\mathbf{N}}$. 
(The routine only checks
LDR if
WANTR is supplied as .TRUE..)
 ${\mathbf{IFAIL}}>1$
The $QR$ algorithm has failed to converge to the singular values in $50\times {\mathbf{N}}$ iterations. In this case ${\mathbf{SV}}\left(1\right),{\mathbf{SV}}\left(2\right),\dots ,{\mathbf{SV}}\left({\mathbf{IFAIL}}1\right)$ may not have been correctly found and the remaining singular values may not be the smallest singular values. The matrix $A$ has nevertheless been factorized as $A={Q}_{1}C{P}^{\mathrm{T}}$, where $C$ is an upper bidiagonal matrix with ${\mathbf{SV}}\left(1\right),{\mathbf{SV}}\left(2\right),\dots ,{\mathbf{SV}}\left(n\right)$ as its diagonal elements and ${\mathbf{WORK}}\left(2\right),{\mathbf{WORK}}\left(3\right),\dots ,{\mathbf{WORK}}\left(n\right)$ as its superdiagonal elements.
This failure cannot occur if
SVD is returned as .FALSE. and in any case is extremely rare.
7 Accuracy
The computed factors
$Q$,
$U$,
$R$,
$D$ and
${P}^{\mathrm{T}}$ satisfy the relations
where
${\Vert E\Vert}_{2}\le {c}_{1}\epsilon {\Vert A\Vert}_{2}$,
${\Vert F\Vert}_{2}\le {c}_{2}\epsilon {\Vert A\Vert}_{2}$,
$\epsilon $ being the machine precision and ${c}_{1}$ and ${c}_{2}$ are modest functions of $m$ and $n$. Note that ${\Vert A\Vert}_{2}=s{v}_{1}$.
The time taken by F02WDF to obtain the Householder $QU$ factorization is approximately proportional to ${n}^{2}\left(3mn\right)$.
The additional time taken to obtain the singular value decomposition is approximately proportional to ${n}^{3}$, where the constant of proportionality depends upon whether or not the orthogonal matrices $R$ and ${P}^{\mathrm{T}}$ are required.
Singular vectors associated with a zero or multiple singular value, are not uniquely determined, even in exact arithmetic, and very different results may be obtained if they are computed on different machines.
Unless otherwise stated in the
Users' Note for your implementation, the routine may be called with the same array for parameters
Z and
SV, in which case, if
SVD is returned as .TRUE., the singular values will overwrite the original contents of
Z; also, if
${\mathbf{WANTPT}}=\mathrm{.FALSE.}$, it may be called with the same array for parameters
R and
PT. However this is not standard Fortran, and may not work on all systems.
This routine is called by the least squares routine
F04JGF.
Following the
$QU$ factorization of
$A$, if
SVD is supplied as
.FALSE., then the condition number of
$U$ given by
is found, where
${\Vert .\Vert}_{F}$ denotes the Frobenius norm, and if
$C\left(U\right)$ is such that
then
$U$ is regarded as singular and the singular values of
$A$ are computed. If this test is not satisfied, then the rank of
$A$ is set to
$n$. Note that if
SVD is supplied as .TRUE. then this test is omitted.
When the singular values are computed, then the rank of
$A$,
$r$, is returned as the largest integer such that
unless
$s{v}_{1}=0$ in which case
$r$ is returned as zero. That is, singular values which satisfy
$s{v}_{i}\le {\mathbf{TOL}}\times s{v}_{1}$ are regarded as negligible because relative perturbations of order
TOL can make such singular values zero.
The
$k$th Householder transformation matrix,
${T}_{k}$, used in the
$QU$ factorization is chosen to introduce the zeros into the
$k$th column and has the form
where
$u$ is an
$\left(mk+1\right)$ element vector.
In place of
$u$ the routine actually computes the vector
$z$ given by
The first element of
$z$ is stored in
${\mathbf{Z}}\left(k\right)$ and the remaining elements of
$z$ are overwritten on the subdiagonal elements of the
$k$th column of
A. The upper triangular matrix
$U$ is overwritten on the
$n$ by
$n$ upper triangular part of
A.
9 Example
This example obtains the rank and the singular value decomposition of the
$6$ by
$4$ matrix
$A$ given by
the value
TOL to be taken as
$5\times {10}^{4}$.
9.1 Program Text
Program Text (f02wdfe.f90)
9.2 Program Data
Program Data (f02wdfe.d)
9.3 Program Results
Program Results (f02wdfe.r)