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Documentation

Building LINALG

fpm build

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

Standard Solution Example

program example    use iso_fortran_env    use linalgimplicit none    ! Local Variablesreal(dp):: a(3,3), b(3)integer(i32):: i, pvt(3)    ! Build the3-by-3 matrix A.    !     |123 |    ! A= |456 |    !     |780 |    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 factorizationcall 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

LU Solution: X =  0.3333 -0.6667  0.0000

Overdetermined System Example

program example    use iso_fortran_env    use linalgimplicit none    ! Local Variablesreal(dp):: a(3,2), b(3)integer(i32):: i    ! Build the3-by-2 matrix A    !     |21 |    ! A= |-31 |    !     |-11 |    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 first2 elementsin b.call solve_least_squares(a, b)    ! Display the resultsprint'(A)',"Least Squares Solution: X ="print'(F9.5)', (b(i), i=1,size(a,2))end program

Least Squares Solution: X =  0.13158 -0.57895

Eigen Analysis Example

! This is an example illustrating the use of the eigenvalue and eigenvector! routinesto solve a free vibration problem of3 masses connected by springs.!!     k1           k2           k3           k4! |-\/\/\-| m1 |-\/\/\-| m2 |-\/\/\-| m3 |-\/\/\-|!! As illustrated above, the system consists of3 masses connected by springs.! Spring k1 and spring k4 connect the end massesto ground.  The equations of! motion for this system are as follows.!! | m100 | |x1"|   | k1+k2  -k2      0  | |x1|   |0|! | 0   m2  0 | |x2"|+ |-k2  k2+k3-k3 | |x2|= |0|! |00   m3| |x3"|   |   0    -k3    k3+k4| |x3|   |0|!! Notice: x1"= the second time derivative of x1.program example    use iso_fortran_env    use linalgimplicit none    ! Define the model parametersreal(dp),parameter:: pi=3.14159265359d0real(dp),parameter:: m1=0.5d0real(dp),parameter:: m2=2.5d0real(dp),parameter:: m3=0.75d0real(dp),parameter:: k1=5.0d6real(dp),parameter:: k2=10.0d6real(dp),parameter:: k3=10.0d6real(dp),parameter:: k4=5.0d6    ! Local Variablesinteger(i32):: i, jreal(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 eigenvectorscall sort(vals, modeShapes)    ! Compute the natural frequency values, andreturn them with units of Hz.      ! Notice, all eigenvalues and eigenvectors arereal 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 doend program

Modal Information:Mode 1: (232.9225 Hz)    -0.718    -1.000    -0.747Mode 2: (749.6189 Hz)    -0.419    -0.164     1.000Mode 3: (923.5669 Hz)     1.000    -0.184     0.179

Sparse Matrix Example

program example    use iso_fortran_env    use linalgimplicit none    ! Local Variablesinteger(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]    ! Convertto sparse (CSRformat)    ! Note, theassignmentoperator is overloadedto allow conversion.    sparse= dense    ! Compute the solutionto the sparse equationscall sparse_direct_solve(sparse, b, x)  ! Results storedin x    !Print the solutionprint"(A)","Sparse Solution:"print*, x    ! Perform a sanity check on the solution    ! Note, matmul is overloadedto allow multiplication with sparse matrices    bc=matmul(sparse, x)print"(A)","Computed RHS:"print*, bcprint"(A)","Original RHS:"print*, b    ! For comparison, solve the dense system via LU decompositioncall lu_factor(dense, ipiv)call solve_lu(dense, ipiv, b)   ! Results storedin bprint"(A)","Dense Solution:"print*, bend program

Sparse Solution:  0.40000000000000002      -0.18750000000000000        2.6666666666666665       0.42500000000000004     Computed RHS:   2.0000000000000000       -1.5000000000000000        8.0000000000000000        1.0000000000000000Original RHS:   2.0000000000000000       -1.5000000000000000        8.0000000000000000        1.0000000000000000Dense Solution:  0.40000000000000002      -0.18750000000000000        2.6666666666666665       0.42499999999999999

External Libraries