A Fortran 2008 edition of LSQR, a conjugate-gradient type method for solving sparse linear equations and sparse least-squares problems.

LSQR can solve linear systems of the form: A * x = b or [A; damp*I]*x = [b; 0]. Where A is a matrix with m rows and n columns, b is an m-vector, and damp is a scalar. The matrix A is intended to be large and sparse, and may be square or rectangular (over-determined or under-determined).

There are two classes in the library that can be used, lsqr_solver (which is more low-level) and lsqr_solver_ez (which has a simpler interface).

To use the lsqr_solver_ez class, you have to provide the matrix A in sparse form, using three arrays: the row indices, column indices, and the nonzero elements. Here is an example:

program main

 use lsqr_kinds
 use lsqr_module, only: lsqr_solver_ez

 implicit none

 ! define a 3x3 dense system to solve:
 integer,parameter :: m = 3 !! number of rows in `A` matrix
 integer,parameter :: n = 3 !! number of columns in `A` matrix
 real(wp),dimension(m),parameter :: b = real([1,2,3],wp)  !! RHS vector
 integer,dimension(m*n),parameter :: icol = [1,1,1,2,2,2,3,3,3] !! col indices of nonzero elements of `A`
 integer,dimension(m*n),parameter :: irow = [1,2,3,1,2,3,1,2,3] !! row indices of nonzero elements of `A`
 real(wp),dimension(m*n),parameter :: a = real([1,4,7,2,5,88,3,66,9],wp)  !! nonzero elements of `A`

 type(lsqr_solver_ez) :: solver  !! main solver class
 real(wp),dimension(n) :: x   !! solution to `A*x = b`
 integer :: istop  !! solver exit code

 call solver%initialize(m,n,a,irow,icol) ! use defaults for other optional inputs
 call solver%solve(b,zero,x,istop)       ! solve the linear system

 write(*,*) 'istop = ', istop
 write(*,'(1P,A,*(E16.6))') 'x = ', x

end program main

The result from this example is:

 istop = 1
 x = 1.242424E+00   -6.060606E-02   -4.040404E-02

A Fortran Package Manager manifest file is included, so that the library and test cases can be compiled with FPM. For example:

fpm build --profile release
fpm test --profile release

To use lsqr within your fpm project, add the following to your fpm.toml file:

[dependencies]
lsqr = { git="https://github.com/jacobwilliams/lsqr.git" }

The latest API documentation can be found here. This was generated from the source code using FORD.

The lsqr source code and related files and documentation are distributed under a permissive free software license (BSD-style).

• LSMR
• LUSOL
• nlesolver-fortran

• The original Fortran 77 version of this algorithm can be found here: https://web.stanford.edu/group/SOL/software/lsqr/ The updated version has been significantly refactored.
• C.C. Paige and M.A. Saunders, LSQR: An algorithm for sparse linear equations and sparse least squares, ACM Transactions on Mathematical Software 8, 1 (March 1982), pp. 43-71.
• C.C. Paige and M.A. Saunders, Algorithm 583, LSQR: Sparse linear equations and least-squares problems, ACM Transactions on Mathematical Software 8, 2 (June 1982), pp. 195-209.
• C.L. Lawson, R.J. Hanson, D.R. Kincaid and F.T. Krogh, Basic linear algebra subprograms for Fortran usage, ACM Transactions on Mathematical Software 5, 3 (Sept 1979), pp. 308-323 and 324-325.