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<HTML>
<HEAD><TITLE>MB02QY - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="MB02QY">MB02QY</A></H2>
<H3>
Minimum-norm solution to a linear least squares problem, given a rank-revealing QR factorization
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To determine the minimum-norm solution to a real linear least
squares problem:
minimize || A * X - B ||,
using the rank-revealing QR factorization of a real general
M-by-N matrix A, computed by SLICOT Library routine MB03OD.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB02QY( M, N, NRHS, RANK, A, LDA, JPVT, B, LDB, TAU,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK
C .. Array Arguments ..
INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), DWORK( * ), TAU( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
</PRE>
<B>Input/Output Parameters</B>
<PRE>
M (input) INTEGER
The number of rows of the matrices A and B. M >= 0.
N (input) INTEGER
The number of columns of the matrix A. N >= 0.
NRHS (input) INTEGER
The number of columns of the matrix B. NRHS >= 0.
RANK (input) INTEGER
The effective rank of A, as returned by SLICOT Library
routine MB03OD. min(M,N) >= RANK >= 0.
A (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry, the leading min(M,N)-by-N upper trapezoidal
part of this array contains the triangular factor R, as
returned by SLICOT Library routine MB03OD. The strict
lower trapezoidal part of A is not referenced.
On exit, if RANK < N, the leading RANK-by-RANK upper
triangular part of this array contains the upper
triangular matrix R of the complete orthogonal
factorization of A, and the submatrix (1:RANK,RANK+1:N)
of this array, with the array TAU, represent the
orthogonal matrix Z (of the complete orthogonal
factorization of A), as a product of RANK elementary
reflectors.
On exit, if RANK = N, this array is unchanged.
LDA INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT (input) INTEGER array, dimension ( N )
The recorded permutations performed by SLICOT Library
routine MB03OD; if JPVT(i) = k, then the i-th column
of A*P was the k-th column of the original matrix A.
B (input/output) DOUBLE PRECISION array, dimension
( LDB, NRHS )
On entry, if NRHS > 0, the leading M-by-NRHS part of
this array must contain the matrix B (corresponding to
the transformed matrix A, returned by SLICOT Library
routine MB03OD).
On exit, if NRHS > 0, the leading N-by-NRHS part of this
array contains the solution matrix X.
If M >= N and RANK = N, the residual sum-of-squares
for the solution in the i-th column is given by the sum
of squares of elements N+1:M in that column.
If NRHS = 0, the array B is not referenced.
LDB INTEGER
The leading dimension of the array B.
LDB >= max(1,M,N), if NRHS > 0.
LDB >= 1, if NRHS = 0.
TAU (output) DOUBLE PRECISION array, dimension ( min(M,N) )
The scalar factors of the elementary reflectors.
If RANK = N, the array TAU is not referenced.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= max( 1, N, NRHS ).
For good performance, LDWORK should sometimes be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The routine uses a QR factorization with column pivoting:
A * P = Q * R = Q * [ R11 R12 ],
[ 0 R22 ]
where R11 is an upper triangular submatrix of estimated rank
RANK, the effective rank of A. The submatrix R22 can be
considered as negligible.
If RANK < N, then R12 is annihilated by orthogonal
transformations from the right, arriving at the complete
orthogonal factorization:
A * P = Q * [ T11 0 ] * Z.
[ 0 0 ]
The minimum-norm solution is then
X = P * Z' [ inv(T11)*Q1'*B ],
[ 0 ]
where Q1 consists of the first RANK columns of Q.
The input data for MB02QY are the transformed matrices Q' * A
(returned by SLICOT Library routine MB03OD) and Q' * B.
Matrix Q is not needed.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method is numerically stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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