MillepedeII
V040803

MINRESQLP solves symmetric systems Ax = b or min Ax  b_2, where the matrix A may be indefinite and/or singular. More...
Functions/Subroutines  
subroutine, public  minresqlp (n, Aprod, b, shift, Msolve, disable, nout, itnlim, rtol, maxxnorm, trancond, Acondlim, x, istop, itn, rnorm, Arnorm, xnorm, Anorm, Acond) 
Solution of linear equation system or least squares problem. More...  
subroutine, public  symortho (a, b, c, s, r) 
SymOrtho: Stable Householder reflection. More...  
MINRESQLP solves symmetric systems Ax = b or min Ax  b_2, where the matrix A may be indefinite and/or singular.
"A" is really (A  shift*I), where shift is an input real scalar.
09 Sep 2013: Version 27  The software for MINRESQLP is provided by SOL, Stanford University under the terms of the OSI Common Public License (CPL) http://www.opensource.org/licenses/cpl1.0.php or the BSD License http://www.opensource.org/licenses/bsdlicense.php  Authors: SouCheng Choi <sctchoi@uchicago.edu> Computation Institute (CI) University of Chicago Chicago, IL 60637, USA Michael Saunders <saunders@stanford.edu> Systems Optimization Laboratory (SOL) Stanford University Stanford, CA 943054026, USA Contributor: Christopher Paige <paige@cs.mcgill.ca> See also: Makefile
subroutine, public minresqlpmodule::minresqlp  (  integer(ip), intent(in)  n, 
Aprod,  
real(dp), dimension(n), intent(in)  b,  
real(dp), intent(in), optional  shift,  
optional  Msolve,  
logical, intent(in), optional  disable,  
integer(ip), intent(in), optional  nout,  
integer(ip), intent(in), optional  itnlim,  
real(dp), intent(in), optional  rtol,  
real(dp), intent(in), optional  maxxnorm,  
real(dp), intent(in), optional  trancond,  
real(dp), intent(in), optional  Acondlim,  
real(dp), dimension(n), intent(out)  x,  
integer(ip), intent(out), optional  istop,  
integer(ip), intent(out), optional  itn,  
real(dp), intent(out), optional  rnorm,  
real(dp), intent(out), optional  Arnorm,  
real(dp), intent(out), optional  xnorm,  
real(dp), intent(out), optional  Anorm,  
real(dp), intent(out), optional  Acond  
) 
Solution of linear equation system or least squares problem.
 MINRESQLP is designed to solve the system of linear equations Ax = b or the leastsquares problem min  Ax  b _2, where A is an n by n symmetric matrix and b is a given vector. The matrix A may be indefinite and/or singular. 1. If A is known to be positive definite, the Conjugate Gradient Method might be preferred, since it requires roughly the same number of iterations as MINRESQLP but less work per iteration. But if a lowaccuracy solution is adequate, MINRESQLP will terminate sooner. 2. If A is indefinite but Ax = b is known to have a solution (e.g. if A is nonsingular), SYMMLQ might be preferred, since it requires roughly the same number of iterations as MINRESQLP but slightly less work per iteration. 3. If A is indefinite and wellconditioned, and Ax = b has a solution, i.e., it is not a leastsquares problem, MINRES might be preferred as it requires the same number of iterations as MINRESQLP but slightly less work per iteration. The matrix A is intended to be large and sparse. It is accessed by means of a subroutine call of the form call Aprod ( n, x, y ) which must return the product y = Ax for any given vector x. More generally, MINRESQLP is designed to solve the system (A  shift*I) x = b or min  (A  shift*I) x  b _2, where shift is a specified real scalar. Again, the matrix (A  shift*I) may be indefinite and/or singular. The work per iteration is very slightly less if shift = 0. Note: If shift is an approximate eigenvalue of A and b is an approximate eigenvector, x might prove to be a better approximate eigenvector, as in the methods of inverse iteration and/or Rayleighquotient iteration. However, we are not yet sure on that  it may be better to use SYMMLQ. In this documentation, ' denotes the transpose of a vector or a matrix. A further option is that of preconditioning, which may reduce the number of iterations required. If M = C C' is a positive definite matrix that is known to approximate (A  shift*I) in some sense, and if systems of the form My = x can be solved efficiently, the parameter Msolve may be used (see below). When an external procedure Msolve is supplied, MINRESQLP will implicitly solve the system of equations P (A  shift*I) P' xbar = P b, i.e. Abar xbar = bbar where P = C**(1), Abar = P (A  shift*I) P', bbar = P b, and return the solution x = P' xbar. The associated residual is rbar = bbar  Abar xbar = P (b  (A  shift*I)x) = P r. In the discussion below, eps refers to the machine precision. eps is computed by MINRESQLP. A typical value is eps = 2.22d16 for IEEE doubleprecision arithmetic. Parameters  Some inputs are optional, with default values described below. Mandatory inputs are n, Aprod, and b. All outputs other than x are optional. n input The dimension of the matrix or operator A. b(n) input The rhs vector b. x(n) output Returns the computed solution x. Aprod external A subroutine defining the matrix A. For a given vector x, the statement call Aprod ( n, x, y ) must return the product y = Ax without altering the vector x. An extra call of Aprod is used to check if A is symmetric. Msolve external An optional subroutine defining a preconditioning matrix M, which should approximate (A  shift*I) in some sense. M must be positive definite. For a given vector x, the statement call Msolve( n, x, y ) must solve the linear system My = x without altering the vector x. In general, M should be chosen so that Abar has clustered eigenvalues. For example, if A is positive definite, Abar would ideally be close to a multiple of I. If A or A  shift*I is indefinite, Abar might be close to a multiple of diag( I I ). shift input Should be zero if the system Ax = b is to be solved. Otherwise, it could be an approximation to an eigenvalue of A, such as the Rayleigh quotient b'Ab / (b'b) corresponding to the vector b. If b is sufficiently like an eigenvector corresponding to an eigenvalue near shift, then the computed x may have very large components. When normalized, x may be closer to an eigenvector than b. Default to 0. nout input A file number. The calling program must open a file for output using for example: open(nout, file='MINRESQLP.txt', status='unknown') If nout > 0, a summary of the iterations will be printed on unit nout. If nout is absent or the file associated with nout is not opened properly, results will be written to 'MINRESQLP_tmp.txt'. (Avoid 0, 5, 6 because by convention stderr=0, stdin=5, stdout=6.) itnlim input An upper limit on the number of iterations. Default to 4n. rtol input A userspecified tolerance. MINRESQLP terminates if it appears that norm(rbar) is smaller than rtol*[norm(Abar)*norm(xbar) + norm(b)], where rbar = bbar  Abar xbar, or that norm(Abar*rbar) is smaller than rtol*norm(Abar)*norm(rbar). If shift = 0 and Msolve is absent, MINRESQLP terminates if norm(r) is smaller than rtol*[norm(A)*norm(x) + norm(b)], where r = b  Ax, or if norm(A*r) is smaller than rtol*norm(A)*norm(r). Default to machine precision. istop output An integer giving the reason for termination... 0 Initial value of istop. 1 beta_{k+1} < eps. Iteration k is the final Lanczos step. 2 beta2 = 0 in the Lanczos iteration; i.e. the second Lanczos vector is zero. This means the rhs is very special. If there is no preconditioner, b is an eigenvector of Abar. Also, x = (1/alpha1) b is a solution of Abar x = b. Otherwise (if Msolve is present), let My = b. If shift is zero, y is a solution of the generalized eigenvalue problem Ay = lambda My, with lambda = alpha1 from the Lanczos vectors. In general, (A  shift*I)x = b has the solution x = (1/alpha1) y where My = b. 3 b = 0, so the exact solution is x = 0. No iterations were performed. 4 Norm(rbar) appears to be less than the value rtol * [ norm(Abar) * norm(xbar) + norm(b) ]. The solution in x should be an acceptable solution of Abar x = b. 5 Norm(rbar) appears to be less than the value eps * norm(Abar) * norm(xbar). This means that the solution is as accurate as seems reasonable on this machine. 6 Norm(Abar rbar) appears to be less than the value rtol * norm(Abar) * norm(rbar). The solution in x should be an acceptable leastsquares solution. 7 Norm(Abar rbar) appears to be less than the value eps * norm(Abar) * norm(rbar). This means that the leastsquares solution is as accurate as seems reasonable on this machine. 8 The iteration limit was reached before convergence. 9 The matrix defined by Aprod does not appear to be symmetric. For certain vectors y = Av and r = Ay, the products y'y and r'v differ significantly. 10 The matrix defined by Msolve does not appear to be symmetric. For vectors satisfying My = v and Mr = y, the products y'y and r'v differ significantly. 11 An inner product of the form x' M**(1) x was not positive, so the preconditioning matrix M does not appear to be positive definite. 12 xnorm has exceeded maxxnorm or will exceed it next iteration. 13 Acond (see below) has exceeded Acondlim or 0.1/eps, so the matrix Abar must be very illconditioned. 14  gamma_k  < eps. This is very likely a leastsquares problem but x may not contain an acceptable solution yet. 15 norm(Abar x) < rtol * norm(Abar) * norm(x). If disable = .true., then a null vector will be obtained, given rtol. If istop >= 7, the final x may not be an acceptable solution. itn output The number of iterations performed. Anorm output An estimate of the norm of the matrix operator Abar = P (A  shift*I) P', where P = C**(1). Acond output An estimate of the condition of Abar above. This will usually be a substantial underestimate of the true condition. rnorm output An estimate of the norm of the final transformed residual vector, P (b  (A  shift*I) x). xnorm output An estimate of the norm of xbar. This is sqrt( x'Mx ). If Msolve is absent, xnorm is an estimate of norm(x). maxxnorm input An upper bound on norm(x). Default value is 1e7. trancond input If trancond > 1, a switch is made from MINRES iterations to MINRESQLP iterations when Acond > trancond. If trancond = 1, all iterations are MINRESQLP iterations. If trancond = Acondlim, all iterations are conventional MINRES iterations (which are slightly cheaper). Default to 1e7. Acondlim input An upper bound on Acond. Default value is 1e15. disable input All stopping conditions are disabled except norm(Ax) / norm(x) < tol. Default to .false..  MINRESQLP is an implementation of the algorithm described in the following references: SouCheng Choi, Iterative Methods for Singular Linear Equations and Least Squares Problems, PhD dissertation, ICME, Stanford University, 2006. SouCheng Choi, Christopher Paige, and Michael Saunders, MINRESQLP: A Krylov subspace method for indefinite or singular symmetric systems, SIAM Journal of Scientific Computing 33:4 (2011) 18101836. SouCheng Choi and Michael Saunders, ALGORITHM & DOCUMENTATION: MINRESQLP for singular Symmetric and Hermitian linear equations and leastsquares problems, Technical Report, ANL/MCSP30270812, Computation Institute, University of Chicago/Argonne National Laboratory, 2012. SouCheng Choi and Michael Saunders, ALGORITHM xxx: MINRESQLP for singular Symmetric and Hermitian linear equations and leastsquares problems, ACM Transactions on Mathematical Software, to appear, 2013. FORTRAN 90 and MATLAB implementations are downloadable from http://www.stanford.edu/group/SOL/software.html http://home.uchicago.edu/sctchoi/  MINRESQLP development: 14 Dec 2006: SouCheng's thesis completed. MINRESQLP includes a stopping rule for singular systems (using an estimate of Ar) and very many other things(!). Note that Ar small => r is a null vector for A. 09 Oct 2007: F90 version constructed from the F77 version. Initially used compiler option r8, but this is nonstandard. 15 Oct 2007: Test on Arnorm = Ar added to recognize singular systems. 15 Oct 2007: Temporarily used real(8) everywhere. 16 Oct 2007: Use minresqlpDataModule to define dp = selected_real_kind(15). We need "use minresqlpDataModule" at the beginning of modules AND inside interfaces. 06 Jun 2010: Added comments. 12 Jul 2011: Created complex version zminresqlpModule.f90 from real version minresqlpModule.f90. 23 Aug 2011: (1) Tim Hopkins ran version 17 on the NAG Fortran compiler We removed half a dozen unused variables in MINRESQLP and also local var sgn_a and sgn_b in SMMORTHO, as they result in division by zero for inputs a=b=0. (2) Version 18 was submitted to ACM TOMS for review. 20 Aug 2012: Version 19: (1) Added optional inputs and outputs, and default values for optional inputs. (2) Removed inputs 'checkA' and 'precon'. (3) Changed slightly the order of parameters in the MINRESQLP API. (4) Updated documentation. (5) Fixed a minor bug in printing x(1) in iteration log during MINRES mode. (6) Made sure MINRESQLP is portable in both single and double precison. (7) Fixed a bug to ensure the 2x2 Hermitian reflectors are orthonormal. Make output c real. 24 Apr 2013: istop = 12 now means xnorm just exceeded maxxnorm. 28 Jun 2013: likeLS introduced to terminate with big xnorm only if the problem seems to be singular and inconsistent. 08 Jul 2013: (1) dot_product replaces ddotc. 04 Aug 2013: If present(maxxnorm), use maxxnorm_ = min(maxxnorm, one/eps). 09 Sep 2013: Initialize relresl and relAresl to zero. 
Definition at line 411 of file minresqlpModule.f90.
References minresqlpdatamodule::debug, minresqlpblasmodule::dnrm2(), minresqlpdatamodule::eps, minresqlpdatamodule::one, minresqlpdatamodule::prcsn, symortho(), and minresqlpdatamodule::zero.
Referenced by mminrsqlp(), and solgloqlp().
subroutine, public minresqlpmodule::symortho  (  real(dp), intent(in)  a, 
real(dp), intent(in)  b,  
real(dp), intent(out)  c,  
real(dp), intent(out)  s,  
real(dp), intent(out)  r  
) 
SymOrtho: Stable Householder reflection.
USAGE: SymOrtho(a, b, c, s, r) INPUTS: a first element of a twovector [a; b] b second element of a twovector [a; b] OUTPUTS: c cosine(theta), where theta is the implicit angle of reflection s sine(theta) r twonorm of [a; b] DESCRIPTION: Stable Householder reflection that gives c and s such that [ c s ][a] = [r], [ s c ][b] [0] where r = two norm of vector [a, b], c = a / sqrt(a**2 + b**2) = a / r, s = b / sqrt(a**2 + b**2) = b / r. The implementation guards against overlow in computing sqrt (a**2 + b**2). REFERENCES: Algorithm 4.9, stable unsymmetric Givens rotations in Golub and van Loan's book Matrix Computations, 3rd edition. MODIFICATION HISTORY: 20/08/2012: Fixed a bug to ensure the 2x2 Hermitian reflectors are orthonormal. 05/27/2011: Created this file from Matlab SymGivens2.m KNOWN BUGS: MM/DD/2004: description AUTHORS: SouCheng Choi, CI, University of Chicago Michael Saunders, MS&E, Stanford University CREATION DATE: 05/27/2011
Definition at line 1334 of file minresqlpModule.f90.
References minresqlpdatamodule::debug, minresqlpdatamodule::one, minresqlpdatamodule::realmin, and minresqlpdatamodule::zero.
Referenced by minresqlp().