subroutine zqrsl(x,ldx,n,k,qraux,y,qy,qty,b,rsd,xb,job,info) integer ldx,n,k,job,info complex*16 x(ldx,1),qraux(1),y(1),qy(1),qty(1),b(1),rsd(1),xb(1) c c zqrsl applies the output of zqrdc to compute coordinate c transformations, projections, and least squares solutions. c for k .le. min(n,p), let xk be the matrix c c xk = (x(jpvt(1)),x(jpvt(2)), ... ,x(jpvt(k))) c c formed from columnns jpvt(1), ... ,jpvt(k) of the original c n x p matrix x that was input to zqrdc (if no pivoting was c done, xk consists of the first k columns of x in their c original order). zqrdc produces a factored unitary matrix q c and an upper triangular matrix r such that c c xk = q * (r) c (0) c c this information is contained in coded form in the arrays c x and qraux. c c on entry c c x complex*16(ldx,p). c x contains the output of zqrdc. c c ldx integer. c ldx is the leading dimension of the array x. c c n integer. c n is the number of rows of the matrix xk. it must c have the same value as n in zqrdc. c c k integer. c k is the number of columns of the matrix xk. k c must nnot be greater than min(n,p), where p is the c same as in the calling sequence to zqrdc. c c qraux complex*16(p). c qraux contains the auxiliary output from zqrdc. c c y complex*16(n) c y contains an n-vector that is to be manipulated c by zqrsl. c c job integer. c job specifies what is to be computed. job has c the decimal expansion abcde, with the following c meaning. c c if a.ne.0, compute qy. c if b,c,d, or e .ne. 0, compute qty. c if c.ne.0, compute b. c if d.ne.0, compute rsd. c if e.ne.0, compute xb. c c note that a request to compute b, rsd, or xb c automatically triggers the computation of qty, for c which an array must be provided in the calling c sequence. c c on return c c qy complex*16(n). c qy conntains q*y, if its computation has been c requested. c c qty complex*16(n). c qty contains ctrans(q)*y, if its computation has c been requested. here ctrans(q) is the conjugate c transpose of the matrix q. c c b complex*16(k) c b contains the solution of the least squares problem c c minimize norm2(y - xk*b), c c if its computation has been requested. (note that c if pivoting was requested in zqrdc, the j-th c component of b will be associated with column jpvt(j) c of the original matrix x that was input into zqrdc.) c c rsd complex*16(n). c rsd contains the least squares residual y - xk*b, c if its computation has been requested. rsd is c also the orthogonal projection of y onto the c orthogonal complement of the column space of xk. c c xb complex*16(n). c xb contains the least squares approximation xk*b, c if its computation has been requested. xb is also c the orthogonal projection of y onto the column space c of x. c c info integer. c info is zero unless the computation of b has c been requested and r is exactly singular. in c this case, info is the index of the first zero c diagonal element of r and b is left unaltered. c c the parameters qy, qty, b, rsd, and xb are not referenced c if their computation is not requested and in this case c can be replaced by dummy variables in the calling program. c to save storage, the user may in some cases use the same c array for different parameters in the calling sequence. a c frequently occuring example is when one wishes to compute c any of b, rsd, or xb and does not need y or qty. in this c case one may identify y, qty, and one of b, rsd, or xb, while c providing separate arrays for anything else that is to be c computed. thus the calling sequence c c call zqrsl(x,ldx,n,k,qraux,y,dum,y,b,y,dum,110,info) c c will result in the computation of b and rsd, with rsd c overwriting y. more generally, each item in the following c list contains groups of permissible identifications for c a single callinng sequence. c c 1. (y,qty,b) (rsd) (xb) (qy) c c 2. (y,qty,rsd) (b) (xb) (qy) c c 3. (y,qty,xb) (b) (rsd) (qy) c c 4. (y,qy) (qty,b) (rsd) (xb) c c 5. (y,qy) (qty,rsd) (b) (xb) c c 6. (y,qy) (qty,xb) (b) (rsd) c c in any group the value returned in the array allocated to c the group corresponds to the last member of the group. c c linpack. this version dated 08/14/78 . c g.w. stewart, university of maryland, argonne national lab. c c zqrsl uses the following functions and subprograms. c c blas zaxpy,zcopy,zdotc c fortran dabs,min0,mod c c internal variables c integer i,j,jj,ju,kp1 complex*16 zdotc,t,temp logical cb,cqy,cqty,cr,cxb c complex*16 zdum double precision cabs1 double precision dreal,dimag complex*16 zdumr,zdumi dreal(zdumr) = zdumr dimag(zdumi) = (0.0d0,-1.0d0)*zdumi cabs1(zdum) = dabs(dreal(zdum)) + dabs(dimag(zdum)) c c set info flag. c info = 0 c c determine what is to be computed. c cqy = job/10000 .ne. 0 cqty = mod(job,10000) .ne. 0 cb = mod(job,1000)/100 .ne. 0 cr = mod(job,100)/10 .ne. 0 cxb = mod(job,10) .ne. 0 ju = min0(k,n-1) c c special action when n=1. c if (ju .ne. 0) go to 40 if (cqy) qy(1) = y(1) if (cqty) qty(1) = y(1) if (cxb) xb(1) = y(1) if (.not.cb) go to 30 if (cabs1(x(1,1)) .ne. 0.0d0) go to 10 info = 1 go to 20 10 continue b(1) = y(1)/x(1,1) 20 continue 30 continue if (cr) rsd(1) = (0.0d0,0.0d0) go to 250 40 continue c c set up to compute qy or qty. c if (cqy) call zcopy(n,y,1,qy,1) if (cqty) call zcopy(n,y,1,qty,1) if (.not.cqy) go to 70 c c compute qy. c do 60 jj = 1, ju j = ju - jj + 1 if (cabs1(qraux(j)) .eq. 0.0d0) go to 50 temp = x(j,j) x(j,j) = qraux(j) t = -zdotc(n-j+1,x(j,j),1,qy(j),1)/x(j,j) call zaxpy(n-j+1,t,x(j,j),1,qy(j),1) x(j,j) = temp 50 continue 60 continue 70 continue if (.not.cqty) go to 100 c c compute ctrans(q)*y. c do 90 j = 1, ju if (cabs1(qraux(j)) .eq. 0.0d0) go to 80 temp = x(j,j) x(j,j) = qraux(j) t = -zdotc(n-j+1,x(j,j),1,qty(j),1)/x(j,j) call zaxpy(n-j+1,t,x(j,j),1,qty(j),1) x(j,j) = temp 80 continue 90 continue 100 continue c c set up to compute b, rsd, or xb. c if (cb) call zcopy(k,qty,1,b,1) kp1 = k + 1 if (cxb) call zcopy(k,qty,1,xb,1) if (cr .and. k .lt. n) call zcopy(n-k,qty(kp1),1,rsd(kp1),1) if (.not.cxb .or. kp1 .gt. n) go to 120 do 110 i = kp1, n xb(i) = (0.0d0,0.0d0) 110 continue 120 continue if (.not.cr) go to 140 do 130 i = 1, k rsd(i) = (0.0d0,0.0d0) 130 continue 140 continue if (.not.cb) go to 190 c c compute b. c do 170 jj = 1, k j = k - jj + 1 if (cabs1(x(j,j)) .ne. 0.0d0) go to 150 info = j c ......exit go to 180 150 continue b(j) = b(j)/x(j,j) if (j .eq. 1) go to 160 t = -b(j) call zaxpy(j-1,t,x(1,j),1,b,1) 160 continue 170 continue 180 continue 190 continue if (.not.cr .and. .not.cxb) go to 240 c c compute rsd or xb as required. c do 230 jj = 1, ju j = ju - jj + 1 if (cabs1(qraux(j)) .eq. 0.0d0) go to 220 temp = x(j,j) x(j,j) = qraux(j) if (.not.cr) go to 200 t = -zdotc(n-j+1,x(j,j),1,rsd(j),1)/x(j,j) call zaxpy(n-j+1,t,x(j,j),1,rsd(j),1) 200 continue if (.not.cxb) go to 210 t = -zdotc(n-j+1,x(j,j),1,xb(j),1)/x(j,j) call zaxpy(n-j+1,t,x(j,j),1,xb(j),1) 210 continue x(j,j) = temp 220 continue 230 continue 240 continue 250 continue return end