CQRSL(3)

SYNOPSIS
SUBROUTINE CQRSL (X, LDX, N, K, QRAUX, Y, QY, QTY, B, RSD, XB, + JOB, INFO)

NAME
cqrsl - Apply the output of CQRDC to compute coordinate transfor- mations, projections, and least squares solutions.

LIBRARY
SLATEC (LINPACK)

CATEGORY
D9, D2C1

TYPE
COMPLEX (SQRSL-S, DQRSL-D, CQRSL-C)

KEYWORDS
LINEAR ALGEBRA, LINPACK, MATRIX, ORTHOGONAL TRIANGULAR, SOLVE

AUTHOR
Stewart, G. W., (U. of Maryland)

DESCRIPTION
CQRSL applies the output of CQRDC to compute coordinate transformations, projections, and least squares solutions. For K .LE. MIN(N,P), let XK be the matrix XK = (X(JVPT(1)),X(JVPT(2)), ... ,X(JVPT(K))) formed from columns JVPT(1), ... ,JVPT(K) of the original N x P matrix X that was input to CQRDC (if no pivoting was done, XK consists of the first K columns of X in their original order). CQRDC produces a factored unitary matrix Q and an upper triangular matrix R such that XK = Q * (R) (0) This information is contained in coded form in the arrays X and QRAUX. On Entry X COMPLEX(LDX,P). X contains the output of CQRDC. LDX INTEGER. LDX is the leading dimension of the array X. N INTEGER. N is the number of rows of the matrix XK. It must have the same value as N in CQRDC. K INTEGER. K is the number of columns of the matrix XK. K must not be greater than (N,P), where P is the same as in the calling sequence to CQRDC. QRAUX COMPLEX(P). QRAUX contains the auxiliary output from CQRDC. Y COMPLEX(N) Y contains an N-vector that is to be manipulated by CQRSL. JOB INTEGER. JOB specifies what is to be computed. JOB has the decimal expansion ABCDE, with the following meaning. If A .NE. 0, compute QY. If B,C,D, or E .NE. 0, compute QTY. If C .NE. 0, compute B. If D .NE. 0, compute RSD . If E .NE. 0, compute XB. Note that a request to compute B, RSD, or XB automatically triggers the computation of QTY, for which an array must be provided in the calling sequence. On Return QY COMPLEX(N). QY contains Q*Y, if its computation has been requested. QTY COMPLEX(N). QTY contains CTRANS(Q)*Y, if its computation has been requested. Here CTRANS(Q) is the conjugate transpose of the matrix Q. B COMPLEX(K) B contains the solution of the least squares problem minimize NORM2(Y - XK*B), if its computation has been requested. (Note that if pivoting was requested in CQRDC, the J-th component of B will be associated with column JVPT(J) of the original matrix X that was input into CQRDC.) RSD COMPLEX(N). RSD contains the least squares residual Y - XK*B, if its computation has been requested. RSD is also the orthogonal projection of Y onto the orthogonal complement of the column space of XK. XB COMPLEX(N). XB contains the least squares approximation XK*B, if its computation has been requested. XB is also the orthogonal projection of Y onto the column space of X. INFO INTEGER. INFO is zero unless the computation of B has been requested and R is exactly singular. In this case, INFO is the index of the first zero diagonal element of R and B is left unaltered. The parameters QY, QTY, B, RSD, and XB are not referenced if their computation is not requested and in this case can be replaced by dummy variables in the calling program. To save storage, the user may in some cases use the same array for different parameters in the calling sequence. A frequently occurring example is when one wishes to compute any of B, RSD, or XB and does not need Y or QTY. In this case one may identify Y, QTY, and one of B, RSD, or XB, while providing separate arrays for anything else that is to be computed. Thus the calling sequence CALL CQRSL(X,LDX,N,K,QRAUX,Y,DUM,Y,B,Y,DUM,110,INFO) will result in the computation of B and RSD, with RSD overwriting Y. More generally, each item in the following list contains groups of permissible identifications for a single calling sequence. 1. (Y,QTY,B) (RSD) (XB) (QY) 2. (Y,QTY,RSD) (B) (XB) (QY) 3. (Y,QTY,XB) (B) (RSD) (QY) 4. (Y,QY) (QTY,B) (RSD) (XB) 5. (Y,QY) (QTY,RSD) (B) (XB) 6. (Y,QY) (QTY,XB) (B) (RSD) In any group the value returned in the array allocated to the group corresponds to the last member of the group.

REFERENCES
J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W. Stewart, LINPACK Users' Guide, SIAM, 1979.

ROUTINES CALLED
CAXPY, CCOPY, CDOTC

REVISION HISTORY (YYMMDD)
780814 DATE WRITTEN
890531 Changed all specific intrinsics to generic. (WRB)
890831 Modified array declarations. (WRB)
890831 REVISION DATE from Version 3.2
891214 Prologue converted to Version 4.0 format. (BAB)
900326 Removed duplicate information from DESCRIPTION section. (WRB)
920501 Reformatted the REFERENCES section. (WRB)