**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.

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(*)

**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.

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.

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'.

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'.

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

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.

[1] 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.

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

This routine is a LAPACK-based modification of QRSOLV from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors.

**Program Text**

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