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|
program dsvd
c
c This example program is intended to illustrate the
c the use of ARPACK to compute the Singular Value Decomposition.
c
c This code shows how to use ARPACK to find a few of the
c largest singular values(sigma) and corresponding right singular
c vectors (v) for the the matrix A by solving the symmetric problem:
c
c (A'*A)*v = sigma*v
c
c where A is an m by n real matrix.
c
c This code may be easily modified to estimate the 2-norm
c condition number largest(sigma)/smallest(sigma) by setting
c which = 'BE' below. This will ask for a few of the smallest
c and a few of the largest singular values simultaneously.
c The condition number could then be estimated by taking
c the ratio of the largest and smallest singular values.
c
c This formulation is appropriate when m .ge. n.
c Reverse the roles of A and A' in the case that m .le. n.
c
c The main points illustrated here are
c
c 1) How to declare sufficient memory to find NEV
c largest singular values of A .
c
c 2) Illustration of the reverse communication interface
c needed to utilize the top level ARPACK routine DSAUPD
c that computes the quantities needed to construct
c the desired singular values and vectors(if requested).
c
c 3) How to extract the desired singular values and vectors
c using the ARPACK routine DSEUPD.
c
c 4) How to construct the left singular vectors U from the
c right singular vectors V to obtain the decomposition
c
c A*V = U*S
c
c where S = diag(sigma_1, sigma_2, ..., sigma_k).
c
c The only thing that must be supplied in order to use this
c routine on your problem is to change the array dimensions
c appropriately, to specify WHICH singular values you want to
c compute and to supply a the matrix-vector products
c
c w <- Ax
c y <- A'w
c
c in place of the calls to AV( ) and ATV( ) respectively below.
c
c Further documentation is available in the header of DSAUPD
c which may be found in the SRC directory.
c
c This codes implements
c
c\Example-1
c ... Suppose we want to solve A'A*v = sigma*v in regular mode,
c where A is derived from the simplest finite difference
c discretization of the 2-dimensional kernel K(s,t)dt where
c
c K(s,t) = s(t-1) if 0 .le. s .le. t .le. 1,
c t(s-1) if 0 .le. t .lt. s .le. 1.
c
c See subroutines AV and ATV for details.
c ... OP = A'*A and B = I.
c ... Assume "call av (n,x,y)" computes y = A*x
c ... Assume "call atv (n,y,w)" computes w = A'*y
c ... Assume exact shifts are used
c ...
c
c\BeginLib
c
c\Routines called:
c dsaupd ARPACK reverse communication interface routine.
c dseupd ARPACK routine that returns Ritz values and (optionally)
c Ritz vectors.
c dnrm2 Level 1 BLAS that computes the norm of a vector.
c daxpy Level 1 BLAS that computes y <- alpha*x+y.
c dscal Level 1 BLAS thst computes x <- x*alpha.
c dcopy Level 1 BLAS thst computes y <- x.
c
c\Author
c Richard Lehoucq
c Danny Sorensen
c Chao Yang
c Dept. of Computational &
c Applied Mathematics
c Rice University
c Houston, Texas
c
c\SCCS Information: @(#)
c FILE: svd.F SID: 2.4 DATE OF SID: 10/17/00 RELEASE: 2
c
c\Remarks
c 1. None
c
c\EndLib
c
c-----------------------------------------------------------------------
c
c %------------------------------------------------------%
c | Storage Declarations: |
c | |
c | It is assumed that A is M by N with M .ge. N. |
c | |
c | The maximum dimensions for all arrays are |
c | set here to accommodate a problem size of |
c | M .le. MAXM and N .le. MAXN |
c | |
c | The NEV right singular vectors will be computed in |
c | the N by NCV array V. |
c | |
c | The NEV left singular vectors will be computed in |
c | the M by NEV array U. |
c | |
c | NEV is the number of singular values requested. |
c | See specifications for ARPACK usage below. |
c | |
c | NCV is the largest number of basis vectors that will |
c | be used in the Implicitly Restarted Arnoldi |
c | Process. Work per major iteration is |
c | proportional to N*NCV*NCV. |
c | |
c | You must set: |
c | |
c | MAXM: Maximum number of rows of the A allowed. |
c | MAXN: Maximum number of columns of the A allowed. |
c | MAXNEV: Maximum NEV allowed |
c | MAXNCV: Maximum NCV allowed |
c %------------------------------------------------------%
c
integer maxm, maxn, maxnev, maxncv, ldv, ldu
parameter (maxm = 500, maxn=250, maxnev=10, maxncv=25,
& ldu = maxm, ldv=maxn )
c
c %--------------%
c | Local Arrays |
c %--------------%
c
Double precision
& v(ldv,maxncv), u(ldu, maxnev),
& workl(maxncv*(maxncv+8)), workd(3*maxn),
& s(maxncv,2), resid(maxn), ax(maxm)
logical select(maxncv)
integer iparam(11), ipntr(11)
c
c %---------------%
c | Local Scalars |
c %---------------%
c
character bmat*1, which*2
integer ido, m, n, nev, ncv, lworkl, info, ierr,
& j, ishfts, maxitr, mode1, nconv
logical rvec
Double precision
& tol, sigma, temp
c
c %------------%
c | Parameters |
c %------------%
c
Double precision
& one, zero
parameter (one = 1.0D+0, zero = 0.0D+0)
c
c %-----------------------------%
c | BLAS & LAPACK routines used |
c %-----------------------------%
c
Double precision
& dnrm2
external dnrm2, daxpy, dcopy, dscal
c
c %-----------------------%
c | Executable Statements |
c %-----------------------%
c
c %-------------------------------------------------%
c | The following include statement and assignments |
c | initiate trace output from the internal |
c | actions of ARPACK. See debug.doc in the |
c | DOCUMENTS directory for usage. Initially, the |
c | most useful information will be a breakdown of |
c | time spent in the various stages of computation |
c | given by setting msaupd = 1. |
c %-------------------------------------------------%
c
include 'debug.h'
ndigit = -3
logfil = 6
msgets = 0
msaitr = 0
msapps = 0
msaupd = 1
msaup2 = 0
mseigt = 0
mseupd = 0
c
c %-------------------------------------------------%
c | The following sets dimensions for this problem. |
c %-------------------------------------------------%
c
m = 500
n = 100
c
c %------------------------------------------------%
c | Specifications for ARPACK usage are set |
c | below: |
c | |
c | 1) NEV = 4 asks for 4 singular values to be |
c | computed. |
c | |
c | 2) NCV = 20 sets the length of the Arnoldi |
c | factorization |
c | |
c | 3) This is a standard problem |
c | (indicated by bmat = 'I') |
c | |
c | 4) Ask for the NEV singular values of |
c | largest magnitude |
c | (indicated by which = 'LM') |
c | See documentation in DSAUPD for the |
c | other options SM, BE. |
c | |
c | Note: NEV and NCV must satisfy the following |
c | conditions: |
c | NEV <= MAXNEV, |
c | NEV + 1 <= NCV <= MAXNCV |
c %------------------------------------------------%
c
nev = 4
ncv = 10
bmat = 'I'
which = 'LM'
c
if ( n .gt. maxn ) then
print *, ' ERROR with _SVD: N is greater than MAXN '
go to 9000
else if ( m .gt. maxm ) then
print *, ' ERROR with _SVD: M is greater than MAXM '
go to 9000
else if ( nev .gt. maxnev ) then
print *, ' ERROR with _SVD: NEV is greater than MAXNEV '
go to 9000
else if ( ncv .gt. maxncv ) then
print *, ' ERROR with _SVD: NCV is greater than MAXNCV '
go to 9000
end if
c
c %-----------------------------------------------------%
c | Specification of stopping rules and initial |
c | conditions before calling DSAUPD |
c | |
c | abs(sigmaC - sigmaT) < TOL*abs(sigmaC) |
c | computed true |
c | |
c | If TOL .le. 0, then TOL <- macheps |
c | (machine precision) is used. |
c | |
c | IDO is the REVERSE COMMUNICATION parameter |
c | used to specify actions to be taken on return |
c | from DSAUPD. (See usage below.) |
c | |
c | It MUST initially be set to 0 before the first |
c | call to DSAUPD. |
c | |
c | INFO on entry specifies starting vector information |
c | and on return indicates error codes |
c | |
c | Initially, setting INFO=0 indicates that a |
c | random starting vector is requested to |
c | start the ARNOLDI iteration. Setting INFO to |
c | a nonzero value on the initial call is used |
c | if you want to specify your own starting |
c | vector (This vector must be placed in RESID.) |
c | |
c | The work array WORKL is used in DSAUPD as |
c | workspace. Its dimension LWORKL is set as |
c | illustrated below. |
c %-----------------------------------------------------%
c
lworkl = ncv*(ncv+8)
tol = zero
info = 0
ido = 0
c
c %---------------------------------------------------%
c | Specification of Algorithm Mode: |
c | |
c | This program uses the exact shift strategy |
c | (indicated by setting IPARAM(1) = 1.) |
c | IPARAM(3) specifies the maximum number of Arnoldi |
c | iterations allowed. Mode 1 of DSAUPD is used |
c | (IPARAM(7) = 1). All these options can be changed |
c | by the user. For details see the documentation in |
c | DSAUPD. |
c %---------------------------------------------------%
c
ishfts = 1
maxitr = n
mode1 = 1
c
iparam(1) = ishfts
c
iparam(3) = maxitr
c
iparam(7) = mode1
c
c %------------------------------------------------%
c | M A I N L O O P (Reverse communication loop) |
c %------------------------------------------------%
c
10 continue
c
c %---------------------------------------------%
c | Repeatedly call the routine DSAUPD and take |
c | actions indicated by parameter IDO until |
c | either convergence is indicated or maxitr |
c | has been exceeded. |
c %---------------------------------------------%
c
call dsaupd ( ido, bmat, n, which, nev, tol, resid,
& ncv, v, ldv, iparam, ipntr, workd, workl,
& lworkl, info )
c
if (ido .eq. -1 .or. ido .eq. 1) then
c
c %---------------------------------------%
c | Perform matrix vector multiplications |
c | w <--- A*x (av()) |
c | y <--- A'*w (atv()) |
c | The user should supply his/her own |
c | matrix vector multiplication routines |
c | here that takes workd(ipntr(1)) as |
c | the input, and returns the result in |
c | workd(ipntr(2)). |
c %---------------------------------------%
c
call av (m, n, workd(ipntr(1)), ax)
call atv (m, n, ax, workd(ipntr(2)))
c
c %-----------------------------------------%
c | L O O P B A C K to call DSAUPD again. |
c %-----------------------------------------%
c
go to 10
c
end if
c
c %----------------------------------------%
c | Either we have convergence or there is |
c | an error. |
c %----------------------------------------%
c
if ( info .lt. 0 ) then
c
c %--------------------------%
c | Error message. Check the |
c | documentation in DSAUPD. |
c %--------------------------%
c
print *, ' '
print *, ' Error with _saupd, info = ', info
print *, ' Check documentation in _saupd '
print *, ' '
c
else
c
c %--------------------------------------------%
c | No fatal errors occurred. |
c | Post-Process using DSEUPD. |
c | |
c | Computed singular values may be extracted. |
c | |
c | Singular vectors may also be computed now |
c | if desired. (indicated by rvec = .true.) |
c | |
c | The routine DSEUPD now called to do this |
c | post processing |
c %--------------------------------------------%
c
rvec = .true.
c
call dseupd ( rvec, 'All', select, s, v, ldv, sigma,
& bmat, n, which, nev, tol, resid, ncv, v, ldv,
& iparam, ipntr, workd, workl, lworkl, ierr )
c
c %-----------------------------------------------%
c | Singular values are returned in the first |
c | column of the two dimensional array S |
c | and the corresponding right singular vectors |
c | are returned in the first NEV columns of the |
c | two dimensional array V as requested here. |
c %-----------------------------------------------%
c
if ( ierr .ne. 0) then
c
c %------------------------------------%
c | Error condition: |
c | Check the documentation of DSEUPD. |
c %------------------------------------%
c
print *, ' '
print *, ' Error with _seupd, info = ', ierr
print *, ' Check the documentation of _seupd. '
print *, ' '
c
else
c
nconv = iparam(5)
do 20 j=1, nconv
c
s(j,1) = sqrt(s(j,1))
c
c %-----------------------------%
c | Compute the left singular |
c | vectors from the formula |
c | |
c | u = Av/sigma |
c | |
c | u should have norm 1 so |
c | divide by norm(Av) instead. |
c %-----------------------------%
c
call av(m, n, v(1,j), ax)
call dcopy(m, ax, 1, u(1,j), 1)
temp = one/dnrm2(m, u(1,j), 1)
call dscal(m, temp, u(1,j), 1)
c
c %---------------------------%
c | |
c | Compute the residual norm |
c | |
c | || A*v - sigma*u || |
c | |
c | for the NCONV accurately |
c | computed singular values |
c | and vectors. (iparam(5) |
c | indicates how many are |
c | accurate to the requested |
c | tolerance). |
c | Store the result in 2nd |
c | column of array S. |
c %---------------------------%
c
call daxpy(m, -s(j,1), u(1,j), 1, ax, 1)
s(j,2) = dnrm2(m, ax, 1)
c
20 continue
c
c %-------------------------------%
c | Display computed residuals |
c %-------------------------------%
c
call dmout(6, nconv, 2, s, maxncv, -6,
& 'Singular values and direct residuals')
end if
c
c %------------------------------------------%
c | Print additional convergence information |
c %------------------------------------------%
c
if ( info .eq. 1) then
print *, ' '
print *, ' Maximum number of iterations reached.'
print *, ' '
else if ( info .eq. 3) then
print *, ' '
print *, ' No shifts could be applied during implicit',
& ' Arnoldi update, try increasing NCV.'
print *, ' '
end if
c
print *, ' '
print *, ' _SVD '
print *, ' ==== '
print *, ' '
print *, ' Size of the matrix is ', n
print *, ' The number of Ritz values requested is ', nev
print *, ' The number of Arnoldi vectors generated',
& ' (NCV) is ', ncv
print *, ' What portion of the spectrum: ', which
print *, ' The number of converged Ritz values is ',
& nconv
print *, ' The number of Implicit Arnoldi update',
& ' iterations taken is ', iparam(3)
print *, ' The number of OP*x is ', iparam(9)
print *, ' The convergence criterion is ', tol
print *, ' '
c
end if
c
c %-------------------------%
c | Done with program dsvd. |
c %-------------------------%
c
9000 continue
c
end
c
c ------------------------------------------------------------------
c matrix vector subroutines
c
c The matrix A is derived from the simplest finite difference
c discretization of the integral operator
c
c f(s) = integral(K(s,t)x(t)dt).
c
c Thus, the matrix A is a discretization of the 2-dimensional kernel
c K(s,t)dt, where
c
c K(s,t) = s(t-1) if 0 .le. s .le. t .le. 1,
c t(s-1) if 0 .le. t .lt. s .le. 1.
c
c Thus A is an m by n matrix with entries
c
c A(i,j) = k*(si)*(tj - 1) if i .le. j,
c k*(tj)*(si - 1) if i .gt. j
c
c where si = i/(m+1) and tj = j/(n+1) and k = 1/(n+1).
c
c-------------------------------------------------------------------
c
subroutine av (m, n, x, w)
c
c computes w <- A*x
c
integer m, n, i, j
Double precision
& x(n), w(m), one, zero, h, k, s, t
parameter ( one = 1.0D+0, zero = 0.0D+0 )
c
h = one / dble(m+1)
k = one / dble(n+1)
do 5 i = 1,m
w(i) = zero
5 continue
t = zero
c
do 30 j = 1,n
t = t+k
s = zero
do 10 i = 1,j
s = s+h
w(i) = w(i) + k*s*(t-one)*x(j)
10 continue
do 20 i = j+1,m
s = s+h
w(i) = w(i) + k*t*(s-one)*x(j)
20 continue
30 continue
c
return
end
c
c-------------------------------------------------------------------
c
subroutine atv (m, n, w, y)
c
c computes y <- A'*w
c
integer m, n, i, j
Double precision
& w(m), y(n), one, zero, h, k, s, t
parameter ( one = 1.0D+0, zero = 0.0D+0 )
c
h = one / dble(m+1)
k = one / dble(n+1)
do 5 i = 1,n
y(i) = zero
5 continue
t = zero
c
do 30 j = 1,n
t = t+k
s = zero
do 10 i = 1,j
s = s+h
y(j) = y(j) + k*s*(t-one)*w(i)
10 continue
do 20 i = j+1,m
s = s+h
y(j) = y(j) + k*t*(s-one)*w(i)
20 continue
30 continue
c
return
end
c
|
