## MB02YD

### Solving the linear system A x = b, D x = 0, D diagonal

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To determine a vector x which solves the system of linear
equations

A*x = b ,     D*x = 0 ,

in the least squares sense, where A is an m-by-n matrix,
D is an n-by-n diagonal matrix, and b is an m-vector.
It is assumed that a QR factorization, with column pivoting, of A
is available, that is, A*P = Q*R, where P is a permutation matrix,
Q has orthogonal columns, and R is an upper triangular matrix
with diagonal elements of nonincreasing magnitude.
The routine needs the full upper triangle of R, the permutation
matrix P, and the first n components of Q'*b (' denotes the
transpose). The system A*x = b, D*x = 0, is then equivalent to

R*z = Q'*b ,  P'*D*P*z = 0 ,                             (1)

where x = P*z. If this system does not have full rank, then a
least squares solution is obtained. On output, MB02YD also
provides an upper triangular matrix S such that

P'*(A'*A + D*D)*P = S'*S .

The system (1) is equivalent to S*z = c , where c contains the
first n components of the vector obtained by applying to
[ (Q'*b)'  0 ]' the transformations which triangularized
[ R'  P'*D*P ]', getting S.

```
Specification
```      SUBROUTINE MB02YD( COND, N, R, LDR, IPVT, DIAG, QTB, RANK, X, TOL,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         COND
INTEGER           INFO, LDR, LDWORK, N, RANK
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
INTEGER           IPVT(*)
DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), X(*)

```
Arguments

Mode Parameters

```  COND    CHARACTER*1
Specifies whether the condition of the matrix S should be
estimated, as follows:
= 'E' :  use incremental condition estimation and store
the numerical rank of S in RANK;
= 'N' :  do not use condition estimation, but check the
diagonal entries of S for zero values;
= 'U' :  use the rank already stored in RANK.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix R.  N >= 0.

R       (input/output) DOUBLE PRECISION array, dimension (LDR, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the full upper triangle is unaltered, and the
strict lower triangle contains the strict upper triangle
(transposed) of the upper triangular matrix S.

LDR     INTEGER
The leading dimension of array R.  LDR >= MAX(1,N).

IPVT    (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
A*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.

DIAG    (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D.

QTB     (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.

RANK    (input or output) INTEGER
On entry, if COND = 'U', this parameter must contain the
(numerical) rank of the matrix S.
On exit, if COND = 'E' or 'N', this parameter contains
the numerical rank of the matrix S, estimated according
to the value of COND.

X       (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system A*x = b, D*x = 0.

```
Tolerances
```  TOL     DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
rank of the matrix S. If the user sets TOL > 0, then the
given value of TOL is used as a lower bound for the
reciprocal condition number;  a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank.  If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS,  is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S, and
the next N elements contain the solution z.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= 4*N, if COND =  'E';
LDWORK >= 2*N, if COND <> 'E'.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  Standard plane rotations are used to annihilate the elements of
the diagonal matrix D, updating the upper triangular matrix R
and the first n elements of the vector Q'*b. A basic least squares
solution is computed.

```
References
```   More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.

```
Numerical Aspects
```                            2
The algorithm requires 0(N ) operations and is backward stable.

```
```  This routine is a LAPACK-based modification of QRSOLV from the
MINPACK package , and with optional condition estimation.
The option COND = 'U' is useful when dealing with several
right-hand side vectors.

```
Example

Program Text

```  None
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Program Data
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Program Results
```  None
```