Generalized MINRES or Generalized LSQR?

Transcription

1 Generalized MINRES or Generalized LSQR? Michael Saunders Systems Optimization Laboratory (SOL) Institute for Computational Mathematics and Engineering (ICME) Stanford University New Frontiers in Numerical Analysis and Scientific Computing on the occasion of Lothar Reichel s 60th birthday and the 20th anniversary of ETNA Department of Mathematical Sciences Kent State University GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

2 Motivation The Golub-Kahan orthogonal bidiagonalization of A R m n gives us freedom to choose 1 starting vector b R m and solve sparse systems Ax b (as in LSQR) But orthogonal tridiagonalization gives us freedom to choose 2 starting vectors b R m and c R n and solve two sparse systems systems Ax b and A T y c (as in USYMQR GMINRES) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

3 Motivation The Golub-Kahan orthogonal bidiagonalization of A R m n gives us freedom to choose 1 starting vector b R m and solve sparse systems Ax b (as in LSQR) But orthogonal tridiagonalization gives us freedom to choose 2 starting vectors b R m and c R n and solve two sparse systems systems Ax b and A T y c (as in USYMQR GMINRES) Reichel and Ye (2008) chose c to speed up the computation of x Golub, Stoll and Wathen (2008) wanted c T x = b T y GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

4 Abstract Given a general matrix A, we can construct orthogonal matrices U, V that reduce A to tridiagonal form: U T AV = T. We can also arrange that the first columns of U and V are proportional to given vectors b and c. For square A, an iterative form of this orthogonal tridiagonalization was given by Saunders, Simon, and Yip (SINUM 1988) and used to solve square systems Ax = b and A T y = c simultaneously. (One of the resulting solvers becomes MINRES when A is symmetric and b = c.) The approach was rediscovered by Reichel and Ye (NLAA 2008) with emphasis on rectangular A and least-squares problems Ax b. The resulting solver was regarded as a generalization of LSQR (although it doesn t become LSQR in any special case). Careful choice of c was shown to improve convergence. In his last year of life, Gene Golub became interested in GLSQR for estimating c T x = b T y without computing x or y (Golub, Stoll, and Wathen (ETNA 2008)). We review the tridiagonalization process and Gene et al. s insight into its true identity. GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

5 Orthogonal matrix reductions Direct: V = product of Householder transformations n n Iterative: V k = ( v 1 v 2... v k ) n k Mostly short-term recurrences GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

6 Tridiagonalization of symmetric A Direct: ( 1 V T ) ( ) ( 0 b T 1 b A 0 x ) x x x = V x x x x x x x x GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

7 Tridiagonalization of symmetric A Direct: ( 1 V T ) ( ) ( 0 b T 1 b A 0 x ) x x x = V x x x x x x x x Iterative: Lanczos process ( b AVk ) = Vk+1 ( βe1 T k+1,k ) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

8 Bidiagonalization of rectangular A Direct: U T ( b A ) ( 1 V ) = x x x x x x x x x GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

9 Bidiagonalization of rectangular A Direct: U T ( b A ) ( 1 V ) = x x x x x x x x x Iterative: Golub-Kahan process ( b AVk ) = Uk+1 ( βe1 B k+1,k ) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

10 Tridiagonalization of rectangular A Direct: 0 x x x x ( ) ( ) ( ) 1 0 c T 1 x x x U T = b A V x x x x x x GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

11 Tridiagonalization of rectangular A Direct: 0 x x x x ( ) ( ) ( ) 1 0 c T 1 x x x U T = b A V x x x x x x Iterative: S-Simon-Yip (1988), Reichel-Ye (2008) ( b AVk ) = Uk+1 ( βe1 T k+1,k ) ( c A T U k ) = Vk+1 ( γe1 T T k,k+1) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

12 MINRES-type solvers based on Lanczos, Arnoldi, Golub-Kahan, orth-tridiag GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

13 MINRES-type solvers for Ax b A Process Solver symmetric Lanczos Paige-S 1975 MINRES Choi-Paige-S 2011 MINRES-QLP rectangular Golub-Kahan Paige-S 1982 LSQR Fong-S 2011 LSMR unsymmetric Arnoldi Saad-Schultz 1986 GMRES unsymmetric orth-tridiag S-Simon-Yip 1988 USYMQR rectangular orth-tridiag Reichel-Ye 2008 GLSQR GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

14 MINRES-type solvers for Ax b A Process Solver symmetric Lanczos Paige-S 1975 MINRES Choi-Paige-S 2011 MINRES-QLP rectangular Golub-Kahan Paige-S 1982 LSQR Fong-S 2011 LSMR square Arnoldi Saad-Schultz 1986 GMRES square orth-tridiag S-Simon-Yip 1988 GMINRES rectangular orth-tridiag Reichel-Ye 2008 GLSQR GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

15 MINRES-type solvers for Ax b A Process Solver symmetric Lanczos Paige-S 1975 MINRES Choi-Paige-S 2011 MINRES-QLP rectangular Golub-Kahan Paige-S 1982 LSQR Fong-S 2011 LSMR square Arnoldi Saad-Schultz 1986 GMRES square orth-tridiag S-Simon-Yip 1988 GMINRES rectangular orth-tridiag Reichel-Ye 2008 GLSQR All these processes produce similar outputs: ( ) ( ) Lanczos b AVk = Vk+1 βe1 T k+1,k ( ) ( ) Golub-Kahan b AVk = Uk+1 βe1 B k+1,k ( ) ( ) orth-tridiag b AVk = Uk+1 βe1 T k+1,k ( and c A T ) ( U k = Vk+1 γe1 Tk,k+1) T GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

16 MINRES-type solvers for Ax b A Process Solver symmetric Lanczos Paige-S 1975 MINRES Choi-Paige-S 2011 MINRES-QLP rectangular Golub-Kahan Paige-S 1982 LSQR Fong-S 2011 LSMR square Arnoldi Saad-Schultz 1986 GMRES square orth-tridiag S-Simon-Yip 1988 GMINRES rectangular orth-tridiag Reichel-Ye 2008 GLSQR All methods: ( b AVk ) = Uk+1 ( βe1 H k ) b AV k w k = U k+1 (βe 1 H k w k ) b AV k w k U k+1 βe 1 H k w k GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

17 MINRES-type solvers for Ax b A Process Solver symmetric Lanczos Paige-S 1975 MINRES Choi-Paige-S 2011 MINRES-QLP rectangular Golub-Kahan Paige-S 1982 LSQR Fong-S 2011 LSMR square Arnoldi Saad-Schultz 1986 GMRES square orth-tridiag S-Simon-Yip 1988 GMINRES rectangular orth-tridiag Reichel-Ye 2008 GLSQR All methods: ( b AVk ) = Uk+1 ( βe1 H k ) b AV k w k = U k+1 (βe 1 H k w k ) b AV k w k U k+1 βe 1 H k w k x k = V k w k where we choose w k from min βe 1 H k w k GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

18 Symmetric methods for unsymmetric Ax b Lanczos on ( ) ( ) I A r A T δ 2 = I x ( ) b gives Golub-Kahan 0 CG-type subproblem gives LSQR MINRES-type subproblem gives LSMR GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

19 Symmetric methods for unsymmetric Ax b Lanczos on ( ) ( ) I A r A T δ 2 = I x ( ) b gives Golub-Kahan 0 CG-type subproblem gives LSQR MINRES-type subproblem gives LSMR ( ) ( ) ( ) A y b Lanczos on A T = (square A) x c is not equivalent to orthogonal tridiagonalization (but seems worth a try!) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

20 Tridiagonalization of general A using orthogonal matrices Some history GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

21 Orthogonal tridiagonalization 1988 Saunders, Simon, and Yip, SINUM 25 Two CG-type methods for unsymmetric linear equations Focus on square A USYMLQ and USYMQR (GSYMMLQ and GMINRES) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

22 Orthogonal tridiagonalization 1988 Saunders, Simon, and Yip, SINUM 25 Two CG-type methods for unsymmetric linear equations Focus on square A USYMLQ and USYMQR (GSYMMLQ and GMINRES) 2008 Reichel and Ye A generalized LSQR algorithm Focus on rectangular A GLSQR GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

23 Orthogonal tridiagonalization 1988 Saunders, Simon, and Yip, SINUM 25 Two CG-type methods for unsymmetric linear equations Focus on square A USYMLQ and USYMQR (GSYMMLQ and GMINRES) 2008 Reichel and Ye A generalized LSQR algorithm Focus on rectangular A GLSQR 2008 Golub, Stoll, and Wathen Approximation of the scattering amplitude Focus on Ax = b, A T y = c and estimation of c T x = b T y without x, y GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

24 Orthogonal tridiagonalization 1988 Saunders, Simon, and Yip, SINUM 25 Two CG-type methods for unsymmetric linear equations Focus on square A USYMLQ and USYMQR (GSYMMLQ and GMINRES) 2008 Reichel and Ye A generalized LSQR algorithm Focus on rectangular A GLSQR 2008 Golub, Stoll, and Wathen Approximation of the scattering amplitude Focus on Ax = b, A T y = c and estimation of c T x = b T y without x, y 2012 Patrick Küschner, Max Planck Institute, Magdeburg Eigenvalues Need to solve Ax = b and A T y = c GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

25 Original motivation (S 1981) CG, SYMMLQ, MINRES work well for symmetric Ax = b GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

26 Original motivation (S 1981) CG, SYMMLQ, MINRES work well for symmetric Ax = b Tridiagonalization of unsymmetric A is no more than twice the work and storage per iteration GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

27 Original motivation (S 1981) CG, SYMMLQ, MINRES work well for symmetric Ax = b Tridiagonalization of unsymmetric A is no more than twice the work and storage per iteration If A is symmetric, we get Lanczos and MINRES etc GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

28 Original motivation (S 1981) CG, SYMMLQ, MINRES work well for symmetric Ax = b Tridiagonalization of unsymmetric A is no more than twice the work and storage per iteration If A is symmetric, we get Lanczos and MINRES etc If A is nearly symmetric, total itns should be not much more (??) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

29 Elizabeth Yip s SIAM conference abstract (1982) CG method for unsymmetric matrices applied to PDE problems We present a CG-type method to solve Ax = b, where A is an arbitrary nonsingular unsymmetric matrix. The algorithm is equivalent to an orthogonal tridiagonalization of A. GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

30 Elizabeth Yip s SIAM conference abstract (1982) CG method for unsymmetric matrices applied to PDE problems We present a CG-type method to solve Ax = b, where A is an arbitrary nonsingular unsymmetric matrix. The algorithm is equivalent to an orthogonal tridiagonalization of A. Each iteration takes more work than the orthogonal bidiagonalization proposed by Golub-Kahan, Paige-Saunders for sparse least squares problems (LSQR). GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

31 Elizabeth Yip s SIAM conference abstract (1982) CG method for unsymmetric matrices applied to PDE problems We present a CG-type method to solve Ax = b, where A is an arbitrary nonsingular unsymmetric matrix. The algorithm is equivalent to an orthogonal tridiagonalization of A. Each iteration takes more work than the orthogonal bidiagonalization proposed by Golub-Kahan, Paige-Saunders for sparse least squares problems (LSQR). We apply a preconditioned version (Fast Poisson) to the difference equation of unsteady transonic flow with small disturbances. (Compared with ORTHOMIN(5)) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

32 Numerical results with orthogonal tridiagonalization GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

33 Numerical results (SSY 1988) B I I B I A = B = tridiag ( 1 δ 4 1+δ ) I B I I B GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

34 Numerical results (SSY 1988) B I I B I A = B = tridiag ( 1 δ 4 1+δ ) I B I I B Megaflops to reach r 10 6 b : δ ORTHOMIN(5) LSQR GMINRES GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

35 Numerical results (SSY 1988) B I I B I A = B = tridiag ( 1 δ 4 1+δ ) I B I I B Megaflops to reach r 10 6 b : δ ORTHOMIN(5) LSQR GMINRES Bottom line: ORTHOMIN sometimes good, can fail. LSQR always better than GMINRES GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

36 Numerical results (Reichel and Ye 2008) Focused on rectangular A and least-squares (Forgot about SSY 1988 and USYMQR hence GLSQR) Three numerical examples (all square!) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

37 Numerical results (Reichel and Ye 2008) Focused on rectangular A and least-squares (Forgot about SSY 1988 and USYMQR hence GLSQR) Three numerical examples (all square!) Remember x 1 v 1 c (since x k = V k w k and c = γv 1 ) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

38 Numerical results (Reichel and Ye 2008) Focused on rectangular A and least-squares (Forgot about SSY 1988 and USYMQR hence GLSQR) Three numerical examples (all square!) Remember x 1 v 1 c (since x k = V k w k and c = γv 1 ) Focused on choice of c stopping early looking at x k = ( x k1 x k2... x kn ) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

39 Numerical results (Reichel and Ye 2008) Example 1 (Fredholm equation) π 0 Discretize to get Aˆx = ˆb, n = 400 κ(s, t)x(t)dt = b(s), 0 s π 2 Solve Ax = b, b ˆb = 10 3 ˆb GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

40 Numerical results (Reichel and Ye 2008) Example 1 (Fredholm equation) π 0 Discretize to get Aˆx = ˆb, n = 400 κ(s, t)x(t)dt = b(s), 0 s π 2 Among {x LSQR k }, x LSQR 3 is closest to ˆx Solve Ax = b, b ˆb = 10 3 ˆb 101 true x 99 x LSQR 3 GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

41 Numerical results (Reichel and Ye 2008) Example 1 (Fredholm equation) π 0 Discretize to get Aˆx = ˆb, n = 400 κ(s, t)x(t)dt = b(s), 0 s π 2 Among {x LSQR k }, x LSQR 3 is closest to ˆx Solve Ax = b, b ˆb = 10 3 ˆb GLSQR: choose c = ( ) T because true x 100c 101 x GLSQR 1 true x 99 x LSQR 3 GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

42 Numerical results (Reichel and Ye 2008) Example 2 (Star cluster) 470 stars, ˆx = pixels, ˆb = Aˆx, n = Solve Ax = b, b ˆb = 10 2 ˆb GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

43 Numerical results (Reichel and Ye 2008) Example 2 (Star cluster) 470 stars, ˆx = pixels, ˆb = Aˆx, n = Solve Ax = b, b ˆb = 10 2 ˆb Choose c = b (because b x) Compare error in x LSQR k and x GLSQR k x k ˆx / ˆx 0.9 for 40 iterations LSQR GLSQR k GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

44 Numerical results (Reichel and Ye 2008) Example 3 (Fredholm equation) 1 0 k(s, t)x(t)dt = exp(s) + (1 e)s 1, 0 s 1 k(s, t) = { s(t 1), t(s 1), s < t s t Discretize to get Aˆx = ˆb, n = 1024 Solve Ax = b, b ˆb = 10 3 ˆb x LSQR 22 has smallest error, but oscillates around ˆx GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

45 Numerical results (Reichel and Ye 2008) Example 3 (Fredholm equation) 1 0 k(s, t)x(t)dt = exp(s) + (1 e)s 1, 0 s 1 k(s, t) = { s(t 1), t(s 1), s < t s t Discretize to get Aˆx = ˆb, n = 1024 Solve Ax = b, b ˆb = 10 3 ˆb x LSQR 22 has smallest error, but oscillates around ˆx Discretize coarsely to get A c x c = b c, n = 4 Prolongate x c to get x prl R 1024 and starting vector c = x prl x GLSQR 4 is very close to ˆx GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

46 Conclusions GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

47 Subspaces Unsymmetric Lanczos generates two Krylov subspaces: U k span{b Ab A 2 b... A k 1 b} V k span{c A T c (A T ) 2 c... (A T ) k 1 c} GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

48 Subspaces Unsymmetric Lanczos generates two Krylov subspaces: U k span{b Ab A 2 b... A k 1 b} V k span{c A T c (A T ) 2 c... (A T ) k 1 c} Orthogonal tridiagonalization generates U 2k span{b AA T b... (AA T ) k 1 b Ac (AA T )Ac...} V 2k span{c A T Ac... (A T A) k 1 c A T b (A T A)A T b...} GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

49 Subspaces Unsymmetric Lanczos generates two Krylov subspaces: U k span{b Ab A 2 b... A k 1 b} V k span{c A T c (A T ) 2 c... (A T ) k 1 c} Orthogonal tridiagonalization generates U 2k span{b AA T b... (AA T ) k 1 b Ac (AA T )Ac...} V 2k span{c A T Ac... (A T A) k 1 c A T b (A T A)A T b...} Reichel and Ye 2008: Richer subspace for ill-posed Ax b (can choose c x) A can be rectangular Check for early termination of {u k } or {v k } sequence GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

50 Functionals c T x = b T y Lu and Darmofal (SISC 2003) use unsymmetric Lanczos with QMR to solve Ax = b and A T y = c simultaneously and to estimate c T x = b T y at a superconvergent rate: c T x k c T x b T y k b T y b Ax k c A T y k σ min (A) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

51 Functionals c T x = b T y Lu and Darmofal (SISC 2003) use unsymmetric Lanczos with QMR to solve Ax = b and A T y = c simultaneously and to estimate c T x = b T y at a superconvergent rate: c T x k c T x b T y k b T y b Ax k c A T y k σ min (A) Golub, Stoll and Wathen (2008) use orthogonal tridiagonalization with GLSQR to do likewise GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

52 Functionals c T x = b T y Lu and Darmofal (SISC 2003) use unsymmetric Lanczos with QMR to solve Ax = b and A T y = c simultaneously and to estimate c T x = b T y at a superconvergent rate: c T x k c T x b T y k b T y b Ax k c A T y k σ min (A) Golub, Stoll and Wathen (2008) use orthogonal tridiagonalization with GLSQR to do likewise Matrices, moments, and quadrature GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

53 Functionals c T x = b T y Lu and Darmofal (SISC 2003) use unsymmetric Lanczos with QMR to solve Ax = b and A T y = c simultaneously and to estimate c T x = b T y at a superconvergent rate: c T x k c T x b T y k b T y b Ax k c A T y k σ min (A) Golub, Stoll and Wathen (2008) use orthogonal tridiagonalization with GLSQR to do likewise Matrices, moments, and quadrature Golub, Minerbo and Saylor 1998 Nine ways to compute the scattering amplitude (1): Estimating c T x iteratively GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

54 Block Lanczos Orthogonal tridiagonalization is equivalent to block Lanczos on A T A with starting block ( c A T b ) Parlett 1987 GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

55 Block Lanczos Orthogonal tridiagonalization is equivalent to block Lanczos on A T A with starting block ( c A T b ) Parlett 1987 ( block Lanczos on A T Golub, Stoll, and Wathen 2008 ) ( A b with starting block c ) GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

56 Block Lanczos Orthogonal tridiagonalization is equivalent to block Lanczos on A T A with starting block ( c A T b ) Parlett 1987 ( block Lanczos on A T Golub, Stoll, and Wathen 2008 ) ( A b with starting block c ) There are two ways of spreading light. To be the candle or the mirror that reflects it. Edith Wharton GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

57 References M. A. Saunders, H. D. Simon, and E. L. Yip (1988). Two conjugate-gradient-type methods for unsymmetric linear equations, SIAM J. Numer. Anal. 25:4, L. Reichel and Q. Ye (2008). A generalized LSQR algorithm, Numer. Linear Algebra Appl. 15, G. H. Golub, M. Stoll, and A. Wathen (2008). Approximation of the scattering amplitude and linear systems, ETNA 31, GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

58 References M. A. Saunders, H. D. Simon, and E. L. Yip (1988). Two conjugate-gradient-type methods for unsymmetric linear equations, SIAM J. Numer. Anal. 25:4, L. Reichel and Q. Ye (2008). A generalized LSQR algorithm, Numer. Linear Algebra Appl. 15, G. H. Golub, M. Stoll, and A. Wathen (2008). Approximation of the scattering amplitude and linear systems, ETNA 31, Special thanks to Chris Paige, Martin Stoll, and Sou-Cheng Choi GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

59 References M. A. Saunders, H. D. Simon, and E. L. Yip (1988). Two conjugate-gradient-type methods for unsymmetric linear equations, SIAM J. Numer. Anal. 25:4, L. Reichel and Q. Ye (2008). A generalized LSQR algorithm, Numer. Linear Algebra Appl. 15, G. H. Golub, M. Stoll, and A. Wathen (2008). Approximation of the scattering amplitude and linear systems, ETNA 31, Special thanks to Chris Paige, Martin Stoll, and Sou-Cheng Choi Happy birthday Lothar! Thanks for noticing A can be rectangular! GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26

60 Orthogonal reductions MINRES solvers Orth-tridiag Results Conclusions Gene is with us every day GMINRES or GLSQR? LR60/ETNA20 Apr 19 20, /26