subroutine rwupdt(n,r,ldr,w,b,alpha,cos,sin)
integer n,ldr
real alpha
real r(ldr,n),w(n),b(n),cos(n),sin(n)
c **********
c
c subroutine rwupdt
c
c given an n by n upper triangular matrix r, this subroutine
c computes the qr decomposition of the matrix formed when a row
c is added to r. if the row is specified by the vector w, then
c rwupdt determines an orthogonal matrix q such that when the
c n+1 by n matrix composed of r augmented by w is premultiplied
c by (q transpose), the resulting matrix is upper trapezoidal.
c the matrix (q transpose) is the product of n transformations
c
c g(n)*g(n-1)* ... *g(1)
c
c where g(i) is a givens rotation in the (i,n+1) plane which
c eliminates elements in the (n+1)-st plane. rwupdt also
c computes the product (q transpose)*c where c is the
c (n+1)-vector (b,alpha). q itself is not accumulated, rather
c the information to recover the g rotations is supplied.
c
c the subroutine statement is
c
c subroutine rwupdt(n,r,ldr,w,b,alpha,cos,sin)
c
c where
c
c n is a positive integer input variable set to the order of r.
c
c r is an n by n array. on input the upper triangular part of
c r must contain the matrix to be updated. on output r
c contains the updated triangular matrix.
c
c ldr is a positive integer input variable not less than n
c which specifies the leading dimension of the array r.
c
c w is an input array of length n which must contain the row
c vector to be added to r.
c
c b is an array of length n. on input b must contain the
c first n elements of the vector c. on output b contains
c the first n elements of the vector (q transpose)*c.
c
c alpha is a variable. on input alpha must contain the
c (n+1)-st element of the vector c. on output alpha contains
c the (n+1)-st element of the vector (q transpose)*c.
c
c cos is an output array of length n which contains the
c cosines of the transforming givens rotations.
c
c sin is an output array of length n which contains the
c sines of the transforming givens rotations.
c
c subprograms called
c
c fortran-supplied ... abs,sqrt
c
c argonne national laboratory. minpack project. march 1980.
c burton s. garbow, dudley v. goetschel, kenneth e. hillstrom,
c jorge j. more
c
c **********
integer i,j,jm1
real cotan,one,p5,p25,rowj,tan,temp,zero
data one,p5,p25,zero /1.0e0,5.0e-1,2.5e-1,0.0e0/
c
do 60 j = 1, n
rowj = w(j)
jm1 = j - 1
c
c apply the previous transformations to
c r(i,j), i=1,2,...,j-1, and to w(j).
c
if (jm1 .lt. 1) go to 20
do 10 i = 1, jm1
temp = cos(i)*r(i,j) + sin(i)*rowj
rowj = -sin(i)*r(i,j) + cos(i)*rowj
r(i,j) = temp
10 continue
20 continue
c
c determine a givens rotation which eliminates w(j).
c
cos(j) = one
sin(j) = zero
if (rowj .eq. zero) go to 50
if (abs(r(j,j)) .ge. abs(rowj)) go to 30
cotan = r(j,j)/rowj
sin(j) = p5/sqrt(p25+p25*cotan**2)
cos(j) = sin(j)*cotan
go to 40
30 continue
tan = rowj/r(j,j)
cos(j) = p5/sqrt(p25+p25*tan**2)
sin(j) = cos(j)*tan
40 continue
c
c apply the current transformation to r(j,j), b(j), and alpha.
c
r(j,j) = cos(j)*r(j,j) + sin(j)*rowj
temp = cos(j)*b(j) + sin(j)*alpha
alpha = -sin(j)*b(j) + cos(j)*alpha
b(j) = temp
50 continue
60 continue
return
c
c last card of subroutine rwupdt.
c
end