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subroutine cwidth ( w,b,nequ,ncols,integs,nbloks, d, x,iflag )
c this program is a variation of the theme in the algorithm bandet1
c by martin and wilkinson (numer.math. 9(1976)279-307). it solves
c the linear system
c a*x = b
c of nequ equations in case a is almost block diagonal with all
c blocks having ncols columns using no more storage than it takes to
c store the interesting part of a . such systems occur in the determ-
c ination of the b-spline coefficients of a spline approximation.
c
c parameters
c w on input, a two-dimensional array of size (nequ,ncols) contain-
c ing the interesting part of the almost block diagonal coeffici-
c ent matrix a (see description and example below). the array
c integs describes the storage scheme.
c on output, w contains the upper triangular factor u of the
c lu factorization of a possibly permuted version of a . in par-
c ticular, the determinant of a could now be found as
c iflag*w(1,1)*w(2,1)* ... * w(nequ,1) .
c b on input, the right side of the linear system, of length nequ.
c the contents of b are changed during execution.
c nequ number of equations in system
c ncols block width, i.e., number of columns in each block.
c integs integer array, of size (2,nequ), describing the block struct-
c ure of a .
c integs(1,i) = no. of rows in block i = nrow
c integs(2,i) = no. of elimination steps in block i
c = overhang over next block = last
c nbloks number of blocks
c d work array, to contain row sizes . if storage is scarce, the
c array x could be used in the calling sequence for d .
c x on output, contains computed solution (if iflag .ne. 0), of
c length nequ .
c iflag on output, integer
c = (-1)**(no.of interchanges during elimination)
c if a is invertible
c = 0 if a is singular
c
c ------ block structure of a ------
c the interesting part of a is taken to consist of nbloks con-
c secutive blocks, with the i-th block made up of nrowi = integs(1,i)
c consecutive rows and ncols consecutive columns of a , and with
c the first lasti = integs(2,i) columns to the left of the next block.
c these blocks are stored consecutively in the workarray w .
c for example, here is an 11th order matrix and its arrangement in
c the workarray w . (the interesting entries of a are indicated by
c their row and column index modulo 10.)
c
c --- a --- --- w ---
c
c nrow1=3
c 11 12 13 14 11 12 13 14
c 21 22 23 24 21 22 23 24
c 31 32 33 34 nrow2=2 31 32 33 34
c last1=2 43 44 45 46 43 44 45 46
c 53 54 55 56 nrow3=3 53 54 55 56
c last2=3 66 67 68 69 66 67 68 69
c 76 77 78 79 76 77 78 79
c 86 87 88 89 nrow4=1 86 87 88 89
c last3=1 97 98 99 90 nrow5=2 97 98 99 90
c last4=1 08 09 00 01 08 09 00 01
c 18 19 10 11 18 19 10 11
c last5=4
c
c for this interpretation of a as an almost block diagonal matrix,
c we have nbloks = 5 , and the integs array is
c
c i= 1 2 3 4 5
c k=
c integs(k,i) = 1 3 2 3 1 2
c 2 2 3 1 1 4
c
c -------- method --------
c gauss elimination with scaled partial pivoting is used, but mult-
c ipliers are n o t s a v e d in order to save storage. rather, the
c right side is operated on during elimination.
c the two parameters
c i p v t e q and l a s t e q
c are used to keep track of the action. ipvteq is the index of the
c variable to be eliminated next, from equations ipvteq+1,...,lasteq,
c using equation ipvteq (possibly after an interchange) as the pivot
c equation. the entries in the pivot column are a l w a y s in column
c 1 of w . this is accomplished by putting the entries in rows
c ipvteq+1,...,lasteq revised by the elimination of the ipvteq-th
c variable one to the left in w . in this way, the columns of the
c equations in a given block (as stored in w ) will be aligned with
c those of the next block at the moment when these next equations be-
c come involved in the elimination process.
c thus, for the above example, the first elimination steps proceed
c as follows.
c
c *11 12 13 14 11 12 13 14 11 12 13 14 11 12 13 14
c *21 22 23 24 *22 23 24 22 23 24 22 23 24
c *31 32 33 34 *32 33 34 *33 34 33 34
c 43 44 45 46 43 44 45 46 *43 44 45 46 *44 45 46 etc.
c 53 54 55 56 53 54 55 56 *53 54 55 56 *54 55 56
c 66 67 68 69 66 67 68 69 66 67 68 69 66 67 68 69
c . . . .
c
c in all other respects, the procedure is standard, including the
c scaled partial pivoting.
c
integer nbloks, ipvtp1, jmax
integer iflag,integs(2,nbloks),ncols,nequ, i,ii,icount,ipvteq
* ,istar,j,lastcl,lasteq,lasti,nexteq,nrowad
real b(nequ),d(nequ),w(nequ,ncols),x(nequ), awi1od,colmax
* ,ratio,sum ,rowmax,temp
iflag = 1
ipvteq = 0
lasteq = 0
c the i-loop runs over the blocks
do 50 i=1,nbloks
c
c the equations for the current block are added to those current-
c ly involved in the elimination process, by increasing lasteq
c by integs(1,i) after the rowsize of these equations has been
c recorded in the array d .
c
nrowad = integs(1,i)
do 10 icount=1,nrowad
nexteq = lasteq + icount
rowmax = 0.
do 5 j=1,ncols
5 rowmax = amax1(rowmax,abs(w(nexteq,j)))
if (rowmax .eq. 0.) go to 999
10 d(nexteq) = rowmax
lasteq = lasteq + nrowad
c
c there will be lasti = integs(2,i) elimination steps before
c the equations in the next block become involved. further,
c l a s t c l records the number of columns involved in the cur-
c rent elimination step. it starts equal to ncols when a block
c first becomes involved and then drops by one after each elim-
c ination step.
c
lastcl = ncols
lasti = integs(2,i)
do 30 icount=1,lasti
ipvteq = ipvteq + 1
if (ipvteq .lt. lasteq) go to 11
if ( abs(w(ipvteq,1))+d(ipvteq) .gt. d(ipvteq) )
* go to 50
go to 999
c
c determine the smallest i s t a r in (ipvteq,lasteq) for
c which abs(w(istar,1))/d(istar) is as large as possible, and
c interchange equations ipvteq and istar in case ipvteq
c .lt. istar .
c
11 colmax = abs(w(ipvteq,1))/d(ipvteq)
istar = ipvteq
ipvtp1 = ipvteq + 1
do 13 ii=ipvtp1,lasteq
awi1od = abs(w(ii,1))/d(ii)
if (awi1od .le. colmax) go to 13
colmax = awi1od
istar = ii
13 continue
if ( abs(w(istar,1))+d(istar) .eq. d(istar) )
* go to 999
if (istar .eq. ipvteq) go to 16
iflag = -iflag
temp = d(istar)
d(istar) = d(ipvteq)
d(ipvteq) = temp
temp = b(istar)
b(istar) = b(ipvteq)
b(ipvteq) = temp
do 14 j=1,lastcl
temp = w(istar,j)
w(istar,j) = w(ipvteq,j)
14 w(ipvteq,j) = temp
c
c subtract the appropriate multiple of equation ipvteq from
c equations ipvteq+1,...,lasteq to make the coefficient of the
c ipvteq-th unknown (presently in column 1 of w ) zero, but
c store the new coefficients in w one to the left from the old.
c
16 do 20 ii=ipvtp1,lasteq
ratio = w(ii,1)/w(ipvteq,1)
do 18 j=2,lastcl
18 w(ii,j-1) = w(ii,j) - ratio*w(ipvteq,j)
w(ii,lastcl) = 0.
20 b(ii) = b(ii) - ratio*b(ipvteq)
30 lastcl = lastcl - 1
50 continue
c
c at this point, w and b contain an upper triangular linear system
c equivalent to the original one, with w(i,j) containing entry
c (i, i-1+j ) of the coefficient matrix. solve this system by backsub-
c stitution, taking into account its block structure.
c
c i-loop over the blocks, in reverse order
i = nbloks
59 lasti = integs(2,i)
jmax = ncols - lasti
do 70 icount=1,lasti
sum = 0.
if (jmax .eq. 0) go to 61
do 60 j=1,jmax
60 sum = sum + x(ipvteq+j)*w(ipvteq,j+1)
61 x(ipvteq) = (b(ipvteq)-sum)/w(ipvteq,1)
jmax = jmax + 1
70 ipvteq = ipvteq - 1
i = i - 1
if (i .gt. 0) go to 59
return
999 iflag = 0
return
end
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