AD_dnSVM

AD_dnSVM is a Fortran Automatic Differentiation library using forward mode for scalars, vectors and matrices. Its features are:

• no limit in terms of the number of independent variables (this number is defined at runtime)
• up to third derivatives

It is similar to Auto_Deriv fortran module: http://www.autodiff.org/?module=Tools&tool=AUTO_DERIV

date: 20/08/2024

Copyright 2021 David Lauvergnat [1]

Originally, it has been developed for Quantum Model Lib (QML): https://github.com/lauvergn/QuantumModelLib

1) Utilisation with scalars

This library enables to get the derivatives (up to third order) of a function of several variables (type(dnS_t)). See the APP/Example_dnS.f90 file.

1a) Variable initializations

Variables need to be initialized. For instance the variables, X, Y, Z

  USE ADdnSVM_m
  IMPLICIT NONE
  !....

  type(dnS_t) :: X,Y,Z
  type(dnS_t) :: f

  X = Variable( Val=HALF, nVar=3, iVar=1, nderiv=1 )
  Y = Variable( Val=ONE,  nVar=3, iVar=2, nderiv=1 )
  Z = Variable( Val=TWO,  nVar=3, iVar=3, nderiv=1 )

X, Y and Z are, respectively, the first, the second and the third variables (iVar) among 3 (nVar). Their derivatives are defined up to the 1st order (nderiv=1). The variable values are set with Val.

Remark: for Y (the second variable), its value is ONE, the three 1st derivatives are [ZERO,ONE,ZERO] and all 2d derivatives are null.

1b) Variable operations

Operations with X, Y and Z: Remarks: X, Y, Z, f, df, gradf are of dnS_t type and gradf is a table.

    f = ONE + cos(X)**2 + sin(Y*Z)

 f value   : 1. + cos(0.5)**2 + sin(2.) = 2.679448580
 f gradient: [-2. * sin(0.5)*cos(0.5), 2. * cos(2.),   cos(2.)] =
             [-0.8414709848,          -0.8322936731, -0.4161468365]

CALL Write_dnS(f,info='f=1.0 + cos(X)**2 + sin(Y*Z), value: 2.67945')

gives:

 f=1.0 + cos(X)**2 + sin(Y*Z), value: 2.67945
 0   derivative            +0.268E+01
 1st derivative  1         -0.841E+00
 1st derivative  2         -0.832E+00
 1st derivative  3         -0.416E+00

One can define a new function from the first derivative with respect to a variable (defined by the ider index) of a function:

 df = deriv(f,ider=1)

Here, df constains df/dX. WARNING: the nderiv of df is reduced by one with respect to nderiv of f.

The gradient is also possible, the result is in a table of dnS. It has the same limitation with nderiv (reduction of nderiv)/

 gradf = grad(f)

1c) Vector initialization and Jacobian matrix

Instead of several variable initializations, one can initialize a vector:

  USE ADdnSVM_m
  IMPLICIT NONE
  !....

  type(dnS_t), allocatable :: VecOld(:),VecNew(:)

  r  = TWO
  th = Pi/3
  VecOld = Variable([r,th], nderiv=2 )                ! VecOld(1) : r, VecOld(2) : th
  VecNew = VecOld(1)*[cos(VecOld(2)),sin(VecOld(2))]  ! [r*cos(th), r*sin(th)]

  JacNewOld = get_Jacobian( VecNew )                  ! JacNewOld(inew,iold)=[ dQinew/dQiold ]

The vector intialization has optional arguments:

• nVar: the total number of the independent variables. It must be larger or equal to the vector size.
• iVar(:): a table with the indices of the independent variables. It size must be equal to the vector size.

  VecOld = Variable([r,th], nVar=3, iVar=[1,2], nderiv=2 )

1d) Special functions/subroutines

• Subroutine set_dnS(S,d0,d1,d2,d3)to set the corresponding derivatives of S (dnS_t). d0 is real (scalar), d1 is a vector of real, d2 is a matrix of real and d3 has a rank 3. d0, d1, d2 and d3 are optional. nderiv is defined accordingly to the presence of d0, d1, d2 or d3.

• Functions get_d0, get_d1, get_d2 and get_d3 to get, respectively, the corresponding derivatives, dnS%d0, dnS%d1(:), dnS%d2(:,:) and dnS%d3(:,:,:).

• Function get_Flatten to get all dnS derivatives (dnS%d0, dnS%d1(:) ...) in a single real vector.

2) Installation

a) With fpm:

fpm build

b) with a makefile:

make all

It will make the library, the executable tests and example. You can change the compiler, Lapack flag, the OpenMP flag, the compiler optimization flag and the default integer either in the makefile or when calling make:

make all FC=ifort OMP=0 OPT=0 LAPACK=0 INT=4
  # FC=ifort to change the compiller to ifort
  # OMP=0/1 to turn off/on the OpenMP fortran flag.
  # OPT=0/1 to turn off/on the fortran optimization.
  # LAPACK=0/1 to turn off/on the lapack use
  # INT=4/8 to change the default integer

The library, libAD_dnSVM_XXX_oppY_ompZ_lapackW_intV.a is created in the main directory and the libAD_dnSVM.a library is linked to it. Remarks :

• XXX is the compiller (gfortran, ifort ...)
• Y is 0 or 1 (opt0 / opt1: compiler optimization)
• Z is 0 or 1 (omp0 / omp1: whitout/with OpenMP)
• W is 0 or 1 (lapack0 / lapack1: whitout/with lapack)
• V is 4 or 8 (int4 / int8)

If needed, the .mod files are in the OBJ/obj_XXX_oppY_ompZ_lapackW_intV directory.

It has been tested with:

• gfortran (14.1.0_2 on macos, 12 on linux)
• ifx/ifort (2023 on linux)
• nagor (7.1 on linux)

3) run the tests

With fpm

    fpm test dnPoly
    fpm test dnS
    fpm test dnVec

or in the Tests directory, run the scripts

    ./run_test_dnS
    ./run_test_dnPoly

or in the Tests directory (with several combinations of OMP, OPT ...), run the script

    ./run_tests

4) run the example

With fpm

    fpm run Exa_dnS

or in the main directory

    ./Exa_dnS.x