A linear algebra library that provides a user-friendly interface to several BLAS and LAPACK routines. The examples below provide an illustration of just how simple it is to perform a few common linear algebra operations. There is also an optional C API that is available as part of this library.
The documentation can be found here.
CMakeThis library can be built using CMake. For instructions see Running CMake.
FPM can also be used to build this library using the provided fpm.toml.
fpm buildThe LINALG library can be used within your FPM project by adding the following to your fpm.toml file.
[dependencies]
linalg = { git = "https://github.com/jchristopherson/linalg" }This example solves a normally defined system of 3 equations of 3 unknowns.
program example
use iso_fortran_env
use linalg
implicit none
! Local Variables
real(dp) :: a(3,3), b(3)
integer(i32) :: i, pvt(3)
! Build the 3-by-3 matrix A.
! | 1 2 3 |
! A = | 4 5 6 |
! | 7 8 0 |
a = reshape( &
[1.0d0, 4.0d0, 7.0d0, 2.0d0, 5.0d0, 8.0d0, 3.0d0, 6.0d0, 0.0d0], &
[3, 3])
! Build the right-hand-side vector B.
! | -1 |
! b = | -2 |
! | -3 |
b = [-1.0d0, -2.0d0, -3.0d0]
! The solution is:
! | 1/3 |
! x = | -2/3 |
! | 0 |
! Compute the LU factorization
call lu_factor(a, pvt)
! Compute the solution. The results overwrite b.
call solve_lu(a, pvt, b)
! Display the results.
print '(A)', "LU Solution: X = "
print '(F8.4)', (b(i), i = 1, size(b))
end programThe above program produces the following output.
LU Solution: X =
0.3333
-0.6667
0.0000
This example solves an overdetermined system of 3 equations of 2 uknowns.
program example
use iso_fortran_env
use linalg
implicit none
! Local Variables
real(dp) :: a(3,2), b(3)
integer(i32) :: i
! Build the 3-by-2 matrix A
! | 2 1 |
! A = |-3 1 |
! |-1 1 |
a = reshape([2.0d0, -3.0d0, -1.0d0, 1.0d0, 1.0d0, 1.0d0], [3, 2])
! Build the right-hand-side vector B.
! |-1 |
! b = |-2 |
! | 1 |
b = [-1.0d0, -2.0d0, 1.0d0]
! The solution is:
! x = [0.13158, -0.57895]**T
! Compute the solution via a least-squares approach. The results overwrite
! the first 2 elements in b.
call solve_least_squares(a, b)
! Display the results
print '(A)', "Least Squares Solution: X = "
print '(F9.5)', (b(i), i = 1, size(a, 2))
end programThe above program produces the following output.
Least Squares Solution: X =
0.13158
-0.57895
This example computes the eigenvalues and eigenvectors of a mechanical system consisting of several masses connected by springs.
! This is an example illustrating the use of the eigenvalue and eigenvector
! routines to solve a free vibration problem of 3 masses connected by springs.
!
! k1 k2 k3 k4
! |-\/\/\-| m1 |-\/\/\-| m2 |-\/\/\-| m3 |-\/\/\-|
!
! As illustrated above, the system consists of 3 masses connected by springs.
! Spring k1 and spring k4 connect the end masses to ground. The equations of
! motion for this system are as follows.
!
! | m1 0 0 | |x1"| | k1+k2 -k2 0 | |x1| |0|
! | 0 m2 0 | |x2"| + | -k2 k2+k3 -k3 | |x2| = |0|
! | 0 0 m3| |x3"| | 0 -k3 k3+k4| |x3| |0|
!
! Notice: x1" = the second time derivative of x1.
program example
use iso_fortran_env
use linalg
implicit none
! Define the model parameters
real(dp), parameter :: pi = 3.14159265359d0
real(dp), parameter :: m1 = 0.5d0
real(dp), parameter :: m2 = 2.5d0
real(dp), parameter :: m3 = 0.75d0
real(dp), parameter :: k1 = 5.0d6
real(dp), parameter :: k2 = 10.0d6
real(dp), parameter :: k3 = 10.0d6
real(dp), parameter :: k4 = 5.0d6
! Local Variables
integer(i32) :: i, j
real(dp) :: m(3,3), k(3,3), natFreq(3)
complex(dp) :: vals(3), modeShapes(3,3)
! Define the mass matrix
m = reshape([m1, 0.0d0, 0.0d0, 0.0d0, m2, 0.0d0, 0.0d0, 0.0d0, m3], [3, 3])
! Define the stiffness matrix
k = reshape([k1 + k2, -k2, 0.0d0, -k2, k2 + k3, -k3, 0.0d0, -k3, k3 + k4], &
[3, 3])
! Compute the eigenvalues and eigenvectors.
call eigen(k, m, vals, vecs = modeShapes)
! Sort the eigenvalues and eigenvectors
call sort(vals, modeShapes)
! Compute the natural frequency values, and return them with units of Hz.
! Notice, all eigenvalues and eigenvectors are real for this example.
natFreq = sqrt(real(vals)) / (2.0d0 * pi)
! Display the natural frequency and mode shape values.
print '(A)', "Modal Information:"
do i = 1, size(natFreq)
print '(AI0AF8.4A)', "Mode ", i, ": (", natFreq(i), " Hz)"
print '(F10.3)', (real(modeShapes(j,i)), j = 1, size(natFreq))
end do
end programThe above program produces the following output.
Modal Information:
Mode 1: (232.9225 Hz)
-0.718
-1.000
-0.747
Mode 2: (749.6189 Hz)
-0.419
-0.164
1.000
Mode 3: (923.5669 Hz)
1.000
-0.184
0.179
The following example solves a sparse system of equations using a direct solver. The solution is compared to the solution of the same system of equations but in dense format for comparison.
program example
use iso_fortran_env
use linalg
implicit none
! Local Variables
integer(int32) :: ipiv(4)
real(real64) :: dense(4, 4), b(4), x(4), bc(4)
type(csr_matrix) :: sparse
! Build the matrices as dense matrices
dense = reshape([ &
5.0d0, 0.0d0, 0.0d0, 0.0d0, &
0.0d0, 8.0d0, 0.0d0, 6.0d0, &
0.0d0, 0.0d0, 3.0d0, 0.0d0, &
0.0d0, 0.0d0, 0.0d0, 5.0d0], [4, 4])
b = [2.0d0, -1.5d0, 8.0d0, 1.0d0]
! Convert to sparse (CSR format)
! Note, the assignment operator is overloaded to allow conversion.
sparse = dense
! Compute the solution to the sparse equations
call sparse_direct_solve(sparse, b, x) ! Results stored in x
! Print the solution
print "(A)", "Sparse Solution:"
print *, x
! Perform a sanity check on the solution
! Note, matmul is overloaded to allow multiplication with sparse matrices
bc = matmul(sparse, x)
print "(A)", "Computed RHS:"
print *, bc
print "(A)", "Original RHS:"
print *, b
! For comparison, solve the dense system via LU decomposition
call lu_factor(dense, ipiv)
call solve_lu(dense, ipiv, b) ! Results stored in b
print "(A)", "Dense Solution:"
print *, b
end programThe above program produces the following output.
Sparse Solution:
0.40000000000000002 -0.18750000000000000 2.6666666666666665 0.42500000000000004
Computed RHS:
2.0000000000000000 -1.5000000000000000 8.0000000000000000 1.0000000000000000
Original RHS:
2.0000000000000000 -1.5000000000000000 8.0000000000000000 1.0000000000000000
Dense Solution:
0.40000000000000002 -0.18750000000000000 2.6666666666666665 0.42499999999999999
Here is a list of external code libraries utilized by this library.
The dependencies do not necessarily have to be installed to be used. The build will initially look for installed items, but if not found, will then download and build the latest version as part of the build process.