Data

Tools

Indexes

Applications

Experiments

Awesome

An open API service indexing awesome lists of open source software.

https://github.com/jchristopherson/linalg

A linear algebra library that provides a user-friendly interface to several BLAS and LAPACK routines.
https://github.com/jchristopherson/linalg

blas cholesky-decomposition compressed-row-storage compressed-sparse-row eigenvalues eigenvectors fortran lapack linear-algebra lu-decomposition qr-decomposition singular-value-decomposition sparse-matrix

Last synced: 3 months ago
JSON representation

A linear algebra library that provides a user-friendly interface to several BLAS and LAPACK routines.

Awesome Lists containing this project

README 

        
# linalg

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.



## Status

![Build Status](https://github.com/jchristopherson/linalg/actions/workflows/cmake.yml/badge.svg)

[![Actions Status](https://github.com/jchristopherson/linalg/workflows/fpm/badge.svg)](https://github.com/jchristopherson/linalg/actions)



## Documentation

The documentation can be found [here](https://jchristopherson.github.io/linalg/).



## Building LINALG

[CMake](https://cmake.org/)This library can be built using CMake.  For instructions see [Running CMake](https://cmake.org/runningcmake/).



[FPM](https://github.com/fortran-lang/fpm) can also be used to build this library using the provided fpm.toml.

```txt

fpm build

```

The LINALG library can be used within your FPM project by adding the following to your fpm.toml file.

```toml

[dependencies]

linalg = { git = "https://github.com/jchristopherson/linalg" }

```



## Standard Solution Example

This example solves a normally defined system of 3 equations of 3 unknowns.



```fortran

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 program

```

The above program produces the following output.

```text

LU Solution: X =

  0.3333

 -0.6667

  0.0000

```



## Overdetermined System Example

This example solves an overdetermined system of 3 equations of 2 uknowns.



```fortran

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 program

```

The above program produces the following output.

```text

Least Squares Solution: X =

  0.13158

 -0.57895

```



## Eigen Analysis Example

This example computes the eigenvalues and eigenvectors of a mechanical system consisting of several masses connected by springs.



```fortran

! 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 program

```

The above program produces the following output.

```text

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

```



## Sparse Matrix Example

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.

```fortran

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 program

```

The above program produces the following output.

```text

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

```



## External Libraries

Here is a list of external code libraries utilized by this library.

- [BLAS](http://www.netlib.org/blas/)

- [LAPACK](http://www.netlib.org/lapack/)

- [QRUpdate](https://sourceforge.net/projects/qrupdate/)

- [FERROR](https://github.com/jchristopherson/ferror)

- [SPARSKIT](https://www-users.cse.umn.edu/~saad/software/SPARSKIT/)



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.