newton.F90 – GVEC

!!matvec with matmul !#define __MATVEC_N(y,Mat,Vec) y=MATMUL(Mat,Vec) !#define __MATVEC_T(y,Mat,Vec) y=MATMUL(Vec,Mat) !#define __PMATVEC_N(fy,y,Mat,Vec) y=fyy+MATMUL(Mat,Vec) !#define __PMATVEC_T(fy,y,Mat,Vec) y=fyy+MATMUL(Vec,Mat) !#define __AMATVEC_N(y,fMat,Mat,Vec) y=fMatMATMUL(Mat,Vec) !#define __AMATVEC_T(y,fMat,Mat,Vec) y=fMatMATMUL(Vec,Mat) !#define __PAMATVEC_N(fy,y,fMat,Mat,Vec) y=fyy+fMatMATMUL(Mat,Vec) !#define __PAMATVEC_T(fy,y,fMat,Mat,Vec) y=fyy+fMatMATMUL(Vec,Mat)

!!#define __GENERICMATVEC(NT,fy,y,fMat,Mat,Vec) CALL DGEMV(NT,SIZE(Mat,1),SIZE(Mat,2),fMat,Mat,SIZE(Mat,1),Vec,1,fy,y,1)

!!matmat with matmul !#define __MATMAT_NN(Y,A,B) Y=MATMUL(A,B) !#define __MATMAT_TN(Y,A,B) Y=MATMUL(TRANSPOSE(A),B) !#define __MATMAT_NT(Y,A,B) Y=MATMUL(A,TRANSPOSE(B)) !#define __MATMAT_TT(Y,A,B) Y=TRANSPOSE(MATMUL(B,A))

!#define __PMATMAT_NN(fy,Y,A,B) Y=fyY+MATMUL(A,B) !#define __PMATMAT_TN(fy,Y,A,B) Y=fyY+MATMUL(TRANSPOSE(A),B) !#define __PMATMAT_NT(fy,Y,A,B) Y=fyY+MATMUL(A,TRANSPOSE(B)) !#define __PMATMAT_TT(fy,Y,A,B) Y=fyY+TRANSPOSE(MATMUL(B,A))

!#define __AMATMAT_NN(Y,fa,A,B) Y=faMATMUL(A,B) !#define __AMATMAT_TN(Y,fa,A,B) Y=faMATMUL(TRANSPOSE(A),B) !#define __AMATMAT_NT(Y,fa,A,B) Y=faMATMUL(A,TRANSPOSE(B)) !#define __AMATMAT_TT(Y,fa,A,B) Y=faTRANSPOSE(MATMUL(B,A))

!#define __PAMATMAT_NN(fy,Y,fa,A,B) Y=fyY+faMATMUL(A,B) !#define __PAMATMAT_TN(fy,Y,fa,A,B) Y=fyY+faMATMUL(TRANSPOSE(A),B) !#define __PAMATMAT_NT(fy,Y,fa,A,B) Y=fyY+faMATMUL(A,TRANSPOSE(B)) !#define __PAMATMAT_TT(fy,Y,fa,A,B) Y=fyY+faTRANSPOSE(MATMUL(B,A))

! GEMM does in general Y = fa A^?B^? + fy Y ! with structure: (m x n) = (m x k) (k x n) ! Y=A B : DGEMM('N','N',m,n,k,fa,Amat ,m, Bmat,k, fy,Y,m) ! Y=A^TB : DGEMM('T','N',m,n,k,fa,Amat ,k, Bmat,k, fy,Y,m) ! Y=A B^T : DGEMM('N','T',m,n,k,fa,Amat ,m, Bmat,n, fy,Y,m) ! Y=A^T*B^T : DGEMM('T','T',m,n,k,fa,Amat ,k, Bmat,n, fy,Y,m)

!#define __GENERICMATMAT_NN(fy,Y,fa,A,B) CALL DGEMM('N','N',SIZE(A,1),SIZE(B,2),SIZE(B,1),fa,A,SIZE(A,1),B,SIZE(B,1),fy,Y,SIZE(A,1)) !#define __GENERICMATMAT_TN(fy,Y,fa,A,B) CALL DGEMM('T','N',SIZE(A,2),SIZE(B,2),SIZE(B,1),fa,A,SIZE(B,1),B,SIZE(B,1),fy,Y,SIZE(A,2)) !#define __GENERICMATMAT_NT(fy,Y,fa,A,B) CALL DGEMM('N','T',SIZE(A,1),SIZE(B,1),SIZE(B,2),fa,A,SIZE(A,1),B,SIZE(B,1),fy,Y,SIZE(A,1)) !#define __GENERICMATMAT_TT(fy,Y,fa,A,B) CALL DGEMM('T','T',SIZE(A,2),SIZE(B,1),SIZE(B,2),fa,A,SIZE(B,2),B,SIZE(B,1),fy,Y,SIZE(A,2))

! SIMPLE INTERFACE FOR DGEMM, specifying nrows/ncols of mat A and nrows/ncols of mat B (for any transpose!) ! GEMM does in general Y = fa A^?B^? + fy Y ! with structure: (m x n) = (m x k) (k x n) ! Y=A B : DGEMM('N','N',m,n,k,fa,Amat ,m, Bmat,k, fy,Y,m) ! Y=A^TB : DGEMM('T','N',m,n,k,fa,Amat ,k, Bmat,k, fy,Y,m) ! Y=A B^T : DGEMM('N','T',m,n,k,fa,Amat ,m, Bmat,n, fy,Y,m) ! Y=A^T*B^T : DGEMM('T','T',m,n,k,fa,Amat ,k, Bmat,n, fy,Y,m)

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Source Code

!===================================================================================================================================
! Copyright (c) 2025 GVEC Contributors, Max Planck Institute for Plasma Physics
! License: MIT
!===================================================================================================================================
#include "defines.FPP"

!===================================================================================================================================
!>
!!# Module **NEWTON**
!!
!! Some simple Newton solvers
!!
!===================================================================================================================================
MODULE MODgvec_Newton
! MODULES
USE MODgvec_Globals, ONLY:wp,UNIT_stdout,abort
IMPLICIT NONE
PUBLIC

INTERFACE NewtonMin1D
  MODULE PROCEDURE NewtonMin1D
END INTERFACE

INTERFACE NewtonRoot1D
  MODULE PROCEDURE NewtonRoot1D
END INTERFACE

INTERFACE NewtonRoot1D_FdF
  MODULE PROCEDURE NewtonRoot1D_FdF
END INTERFACE

INTERFACE NewtonMin2D
  MODULE PROCEDURE NewtonMin2D
END INTERFACE

INTERFACE NewtonRoot2D
  MODULE PROCEDURE NewtonRoot2D
END INTERFACE

TYPE, ABSTRACT :: c_newton_Min1D
  CONTAINS
  PROCEDURE(i_newton_Min1D), DEFERRED :: FR
  PROCEDURE(i_newton_Min1D), DEFERRED :: dFR
  PROCEDURE(i_newton_Min1D), DEFERRED :: ddFR
END TYPE c_newton_Min1D

TYPE, ABSTRACT :: c_newton_Root1D
  CONTAINS
  PROCEDURE(i_newton_Root1D), DEFERRED :: FR
  PROCEDURE(i_newton_Root1D), DEFERRED :: dFR
END TYPE c_newton_Root1D

!===================================================================================================================================
!> This functional wraps a functional passed to Min1D for use with Root1D (returns the derivative: FR -> dFR, dFR -> ddFR)
!!
!===================================================================================================================================
TYPE, EXTENDS(c_newton_Root1D) :: t_newton_Root1D_wrap_Min1D
  CLASS(c_newton_Min1D), ALLOCATABLE :: parent
  CONTAINS
  PROCEDURE :: FR => newton_Root1D_wrap_Min1D_FR
  PROCEDURE :: dFR => newton_Root1D_wrap_Min1D_dFR
END TYPE t_newton_Root1D_wrap_Min1D

INTERFACE t_newton_Root1D_wrap_Min1D
    MODULE PROCEDURE newton_Root1D_wrap_Min1D_new
END INTERFACE t_newton_Root1D_wrap_Min1D

TYPE, ABSTRACT :: c_newton_Root1D_FdF
  CONTAINS
  PROCEDURE(i_newton_Root1D_FdF), DEFERRED :: FRdFR
END TYPE c_newton_Root1D_FdF

TYPE, ABSTRACT :: c_newton_Min2D
  CONTAINS
  PROCEDURE(i_newton_Min2D_FR), DEFERRED :: FR
  PROCEDURE(i_newton_Min2D_dFR), DEFERRED :: dFR
  PROCEDURE(i_newton_Min2D_ddFR), DEFERRED :: ddFR
END TYPE c_newton_Min2D

TYPE, ABSTRACT :: c_newton_Root2D
  CONTAINS
  PROCEDURE(i_newton_Root2D_FR), DEFERRED :: FR
  PROCEDURE(i_newton_Root2D_dFR), DEFERRED :: dFR
END TYPE c_newton_Root2D

ABSTRACT INTERFACE
  FUNCTION i_newton_Min1D(sf, x) RESULT (y1x1)
    IMPORT wp, c_newton_Min1D
    IMPLICIT NONE
    CLASS(c_newton_Min1D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x
    REAL(wp) :: y1x1
  END FUNCTION i_newton_Min1D

  FUNCTION i_newton_Root1D(sf, x) RESULT (y1x1)
    IMPORT wp, c_newton_Root1D
    IMPLICIT NONE
    CLASS(c_newton_Root1D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x
    REAL(wp) :: y1x1
  END FUNCTION i_newton_Root1D

  FUNCTION i_newton_Root1D_FdF(sf, x) RESULT (y2x1)
    IMPORT wp, c_newton_Root1D_FdF
    IMPLICIT NONE
    CLASS(c_newton_Root1D_FdF), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x
    REAL(wp) :: y2x1(2)
  END FUNCTION i_newton_Root1D_FdF

  FUNCTION i_newton_Min2D_FR(sf, x) RESULT (y1x2)
    IMPORT wp, c_newton_Min2D
    IMPLICIT NONE
    CLASS(c_newton_Min2D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x(2)
    REAL(wp) :: y1x2
  END FUNCTION i_newton_Min2D_FR

  FUNCTION i_newton_Min2D_dFR(sf, x) RESULT (y2x2)
    IMPORT wp, c_newton_Min2D
    IMPLICIT NONE
    CLASS(c_newton_Min2D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x(2)
    REAL(wp) :: y2x2(2)
  END FUNCTION i_newton_Min2D_dFR

  FUNCTION i_newton_Min2D_ddFR(sf, x) RESULT (y22x2)
    IMPORT wp, c_newton_Min2D
    IMPLICIT NONE
    CLASS(c_newton_Min2D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x(2)
    REAL(wp) :: y22x2(2,2)
  END FUNCTION i_newton_Min2D_ddFR

  FUNCTION i_newton_Root2D_FR(sf, x) RESULT (y2x2)
    IMPORT wp, c_newton_Root2D
    IMPLICIT NONE
    CLASS(c_newton_Root2D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x(2)
    REAL(wp) :: y2x2(2)
  END FUNCTION i_newton_Root2D_FR

  FUNCTION i_newton_Root2D_dFR(sf, x) RESULT (y22x2)
    IMPORT wp, c_newton_Root2D
    IMPLICIT NONE
    CLASS(c_newton_Root2D), INTENT(IN) :: sf
    REAL(wp), INTENT(IN) :: x(2)
    REAL(wp) :: y22x2(2, 2)
  END FUNCTION i_newton_Root2D_dFR
END INTERFACE

!===================================================================================================================================

CONTAINS

!===================================================================================================================================
!> Newton's iterative algorithm to find the minimimum of function f(x) in the interval [a,b], using df(x)=0 and the derivative
!!
!===================================================================================================================================
FUNCTION NewtonMin1D(tol,a,b,maxstep,x,fobj) RESULT (fmin)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  REAL(wp),INTENT(IN)               :: tol  !! abort tolerance
  REAL(wp),INTENT(IN)               :: a,b  !! search interval
  REAL(wp),INTENT(IN)               :: maxstep  !! max|dx| allowed
  CLASS(c_newton_Min1D),INTENT(IN)  :: fobj !! functional to minimize with f(x), d/dx f(x), d^2/dx^2 f(x)
  ! OUTPUT VARIABLES -------------------------!
  REAL(wp),INTENT(INOUT) :: x    !! initial guess on input, result on output
  REAL(wp)               :: fmin !! on output =f(x)
  ! LOCAL VARIABLES --------------------------!
  REAL(wp)                                      :: x0
  TYPE(t_newton_Root1D_wrap_Min1D), ALLOCATABLE :: fobj_wrap
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  fobj_wrap = t_newton_Root1D_wrap_Min1D(fobj)
  x0=x
  x=NewtonRoot1D(tol,a,b,maxstep,x0,0.0_wp,fobj_wrap)
  fmin=fobj%FR(x)
  DEALLOCATE(fobj_wrap)
END FUNCTION NewtonMin1D


!===================================================================================================================================
!> constructor for the Min1D type wrapped for Root1D
!!
!===================================================================================================================================
FUNCTION newton_Root1D_wrap_Min1D_new(parent) RESULT(sf)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  CLASS(c_newton_Min1D), INTENT(IN) :: parent
  ! OUTPUT VARIABLES ------------------------!
  TYPE(t_newton_Root1D_wrap_Min1D) :: sf
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  sf%parent = parent
END FUNCTION


!===================================================================================================================================
!> f(x) function of the Min1D type wrapped for Root1D
!!
!===================================================================================================================================
FUNCTION newton_Root1D_wrap_Min1D_FR(sf, x) RESULT(y1x1)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  CLASS(t_newton_Root1D_wrap_Min1D), INTENT(IN) :: sf
  REAL(wp), INTENT(IN) :: x
  ! OUTPUT VARIABLES -------------------------!
  REAL(wp) :: y1x1
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  y1x1 = sf%parent%dFR(x)
END FUNCTION


!===================================================================================================================================
!> d/dx f(x) function of the Min1D type wrapped for Root1D
!!
!===================================================================================================================================
FUNCTION newton_Root1D_wrap_Min1D_dFR(sf, x) RESULT(y1x1)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  CLASS(t_newton_Root1D_wrap_Min1D), INTENT(IN) :: sf
  REAL(wp), INTENT(IN) :: x
  ! OUTPUT VARIABLES -------------------------!
  REAL(wp) :: y1x1
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  y1x1 = sf%parent%ddFR(x)
END FUNCTION


!===================================================================================================================================
!> Newton's iterative algorithm to find the root of function FR(x(:)) in the interval [a(:),b(:)], using d/dx(:)F(x)=0 and the derivative
!!
!===================================================================================================================================
FUNCTION NewtonRoot1D(tol,a,b,maxstep,xin,F0,fobj) RESULT (xout)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  REAL(wp),INTENT(IN) :: tol    !! abort tolerance
  REAL(wp),INTENT(IN) :: a,b    !! search interval
  REAL(wp),INTENT(IN) :: maxstep !! max|dx| allowed
  REAL(wp),INTENT(IN) :: xin    !! initial guess
  REAL(wp),INTENT(IN) :: F0     !! function to find root is FR(x)-F0
  ! OUTPUT VARIABLES -------------------------!
  CLASS(c_newton_Root1D) :: fobj
  REAL(wp)            :: xout    !! on output =f(x)
  ! LOCAL VARIABLES --------------------------!
  INTEGER             :: iter,maxiter
  REAL(wp)            :: x,dx
  LOGICAL             :: converged
  LOGICAL             :: converged2
  ! CODE --------------------------------------------------------------------------------------------------------------------------!

  converged=.FALSE.
  x=xin
  maxiter=20
  DO iter=1,maxiter
    dx=-(fobj%FR(x)-F0)/fobj%dFR(x)
    dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
    x = x+dx
    IF(ABS(dx).GT.maxstep) dx=dx/ABS(dx)*maxstep
    converged=(ABS(dx).LT.tol).AND.(x.GT.a).AND.(x.LT.b)
    IF(converged) EXIT
  END DO !iter
  IF(.NOT.converged) THEN
    !repeat with maxstep /10 and a little change in the initial condition
    x=MIN(b,MAX(a,xin+0.01_wp*(b-a)))
    maxiter=200
    DO iter=1,maxiter
      dx=-(fobj%FR(x)-F0)/fobj%dFR(x)
      dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
      IF(ABS(dx).GT.maxstep) dx=dx/ABS(dx)*0.1_wp*maxstep
      x = x+dx
      converged2=(ABS(dx).LT.tol).AND.(x.GT.a).AND.(x.LT.b)
      IF(converged2) EXIT
    END DO !iter
    IF(converged2) THEN
      xout=x
      RETURN
    END IF
    WRITE(UNIT_stdout,*)'Newton abs(dx)<tol',ABS(dx),tol,ABS(dx).LT.tol
    WRITE(UNIT_stdout,*)'Newton x>a',x,a,(x.GT.a)
    WRITE(UNIT_stdout,*)'Newton x<b',x,b,(x.LT.b)
    WRITE(UNIT_stdout,*)'after iter',iter-1
    CALL abort(__STAMP__, &
              'NewtonRoot1D not converged')
  END IF
  xout=x

END FUNCTION NewtonRoot1D


!===================================================================================================================================
!> Newton's iterative algorithm to find the root of function FR(x(:)) in the interval [a(:),b(:)], using d/dx(:)F(x)=0 and the derivative
!!
!===================================================================================================================================
FUNCTION NewtonRoot1D_FdF(tol,a,b,maxstep,xin,F0,fobj) RESULT (xout)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  REAL(wp),INTENT(IN) :: tol     !! abort tolerance
  REAL(wp),INTENT(IN) :: a,b     !! search interval
  REAL(wp),INTENT(IN) :: maxstep !! max|dx| allowed
  REAL(wp),INTENT(IN) :: xin     !! initial guess on input
  REAL(wp),INTENT(IN) :: F0      !! function to find root is FR(x)-F0
  ! OUTPUT VARIABLES -------------------------!
  CLASS(c_newton_Root1D_FdF) :: fobj !! function to find root f(x) & derivative d/dx f(x) as FRdFR method
  REAL(wp)            :: xout    !! output x for f(x)=0
  ! LOCAL VARIABLES --------------------------!
  INTEGER             :: iter,maxiter
  REAL(wp)            :: x,dx
  REAL(wp)            :: FRdFRx(2) !1: FR(x), 2: dFR(x)
  LOGICAL             :: converged
  LOGICAL             :: converged2
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  converged=.FALSE.
  x=xin
  maxiter=20
  DO iter=1,maxiter
    FRdFRx=fobj%FRdFR(x)
    dx=-(FRdFRx(1)-F0)/FRdFRx(2)
    dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
    IF(ABS(dx).GT.maxstep) dx=dx/ABS(dx)*maxstep
    x = x+dx
    converged=(ABS(dx).LT.tol).AND.(x.GE.a).AND.(x.LE.b)
    IF(converged) EXIT
  END DO !iter
  IF(.NOT.converged) THEN
    !repeat with maxstep /10 and a little change in the initial condition
    converged2=.FALSE.
    x=MIN(b,MAX(a,xin+0.01_wp*(b-a)))
    maxiter=200
    DO iter=1,maxiter
      FRdFRx=fobj%FRdFR(x)
      dx=-(FRdFRx(1)-F0)/FRdFRx(2)
      dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
      IF(ABS(dx).GT.maxstep) dx=dx/ABS(dx)*0.1_wp*maxstep
      x = x+dx
      converged2=(ABS(dx).LT.tol).AND.(x.GE.a).AND.(x.LE.b)
      IF(converged2) EXIT
    END DO !iter
    IF(converged2) THEN
      xout=x
      RETURN
    END IF
    WRITE(UNIT_stdout,*)'Newton abs(dx)<tol',ABS(dx),tol,ABS(dx).LT.tol
    WRITE(UNIT_stdout,*)'Newton x>a',x,a,(x.GT.a)
    WRITE(UNIT_stdout,*)'Newton x<b',x,b,(x.LT.b)
    WRITE(UNIT_stdout,*)'after iter',iter-1
    CALL abort(__STAMP__,&
              'NewtonRoot1D_FdF not converged')
  END IF
  xout=x

END FUNCTION NewtonRoot1D_FdF


!===================================================================================================================================
!> Newton's iterative algorithm to find the minimimum of function f(x,y) in the interval x(i)[a(i),b(i)],
!! using grad(f(x)=0 and the derivative
!!
!===================================================================================================================================
FUNCTION NewtonMin2D(tol,a,b,x,fobj) RESULT (fmin)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  REAL(wp),INTENT(IN)    :: tol        !! abort tolerance
  REAL(wp),INTENT(IN)    :: a(2),b(2)  !! search interval (2D)
  ! OUTPUT VARIABLES -------------------------!
  REAL(wp),INTENT(INOUT) :: x(2) !! initial guess on input, result on output
  CLASS(c_newton_Min2D)  :: fobj !! functional f(x,y) to minimize with f, f', f''
  REAL(wp)               :: fmin !! on output =f(x,y)
  ! LOCAL VARIABLES --------------------------!
  INTEGER             :: iter,maxiter
  REAL(wp)            :: dx(2)
  REAL(wp)            :: det_Hess
  REAL(wp)            :: gradF(2),Hess(2,2),HessInv(2,2)
  LOGICAL             :: converged
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  converged=.FALSE.
  maxiter=50
  DO iter=1,maxiter
    Hess=fobj%ddFR(x)
    det_Hess = Hess(1,1)*Hess(2,2)-Hess(1,2)*Hess(2,1)
    IF(det_Hess.LT.1.0E-12) CALL abort(__STAMP__,&
                                      'det Hessian=0 in NewtonMin')
    HessInv(1,1)= Hess(2,2)
    HessInv(1,2)=-Hess(1,2)
    HessInv(2,1)=-Hess(2,1)
    HessInv(2,2)= Hess(1,1)
    HessInv=HessInv/det_Hess
    gradF=fobj%dFR(x)
    dx=-MATMUL(HessInv,gradF)
    dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
    x = x+dx
    converged=(SQRT(SUM(dx*dx)).LT.tol).AND.ALL(x.GT.a).AND.ALL(x.LT.b)
    IF(converged) EXIT
  END DO !iter
  IF(.NOT.converged) CALL abort(__STAMP__,&
                                'NewtonMin2D not converged')
  fmin=fobj%FR(x)

END FUNCTION NewtonMin2D

!===================================================================================================================================
!> Newton's iterative algorithm to find the root of function [f1(x1,x2),f2(x1,x2)]=[0,0] in the interval a(i)<=x(i)<=b(i),
!! using the Jacobian  dfi/dxj, i=1,2, j=1,2, such that fi(x1,x2)=fi(x1_0,x2_0)+  [dfi/dx1,dfi/dx2].[dx1,dx2]
!! in each step, we find dx1,dx2 st -[[dfi/dxj]] dxj =fi(x1_0,x2_0)
!!
!===================================================================================================================================
FUNCTION NewtonRoot2D(tol,a,b,maxstep,xin,fobj) RESULT (xout)
  ! MODULES
  IMPLICIT NONE
  ! INPUT VARIABLES -------------------------!
  REAL(wp),INTENT(IN)    :: tol        !! abort tolerance
  REAL(wp),INTENT(IN)    :: a(2),b(2)  !! search interval (2D)
  REAL(wp),INTENT(IN) :: maxstep(2) !! max|dx| allowed
  REAL(wp),INTENT(IN)    :: xin(2) !! initial guess
  ! OUTPUT VARIABLES -------------------------!
  CLASS(c_newton_Root2D) :: fobj !! function to find root f1(x1,x2)=0,f2(x1,x2)=0 and derivatives d fi(x1,x2)/dxj
  REAL(wp)               :: xout(2) !! x1,x2 that have f1(x1,x2)=0 and f2(x1,x2)=0
  ! LOCAL VARIABLES --------------------------!
  INTEGER             :: iter,maxiter
  REAL(wp)            :: x(2),dx(2)
  REAL(wp)            :: det_Jac
  REAL(wp)            :: F(2),Jac(2,2),JacInv(2,2)
  LOGICAL             :: converged
  ! CODE --------------------------------------------------------------------------------------------------------------------------!
  converged=.FALSE.
  x=xin
  maxiter=50
  DO iter=1,maxiter
    Jac=fobj%dFR(x)
    det_Jac = Jac(1,1)*Jac(2,2)-Jac(1,2)*Jac(2,1)
    IF(det_Jac.LT.1.0E-12) CALL abort(__STAMP__,&
                                      'det Jacobian<=0 in NewtonRoot2d')
    JacInv(1,1)= Jac(2,2)
    JacInv(1,2)=-Jac(1,2)
    JacInv(2,1)=-Jac(2,1)
    JacInv(2,2)= Jac(1,1)
    JacInv=JacInv/det_Jac
    F=fobj%FR(x)
    dx=-MATMUL(JacInv,F)
    dx = MAX(-(x-a),MIN(b-x,dx)) !respect bounds
    IF(ABS(dx(1)).GT.maxstep(1)) dx(1)=dx(1)/ABS(dx(1))*maxstep(1)
    IF(ABS(dx(2)).GT.maxstep(2)) dx(2)=dx(2)/ABS(dx(2))*maxstep(2)

    x = x+dx
    converged=(SQRT(SUM(dx*dx)).LT.tol).AND.ALL(x.GT.a).AND.ALL(x.LT.b)
    IF(converged) EXIT
  END DO !iter
  xout=x
  IF(.NOT.converged) THEN
    WRITE(UNIT_stdout,*)'Newton abs(dx)<tol',ABS(dx),tol,'F(x)',fobj%FR(xout)
    CALL abort(__STAMP__,&
              'NewtonRoot2D not converged')
  END IF

END FUNCTION NewtonRoot2D

END MODULE MODgvec_Newton

newton.F90 Source File

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