      SUBROUTINE DUMKA3(N,T,TEND,H0,FUN,Y,Z,
     &  Z1,Y_EN,Z_EN,OLDEIGENVECTOR,ATOL,RTOL,IWORK)
 
c ------- ABSTRACT -----------------------------------
c DUMKA3 - integrates initial value problems for systems of first 
c order ordinary differential equations y'=f(y,t). 
c It is based on a family of explicit Runge-Kutta-Chebyshev formulas 
c of order three.
c It uses optimal third order accuracy stability polynomials 
c with the largest stability region along the negative real axis.
c
c
c The algorithm dumka3 and dumka4 is described in the article:
c Medovikov A.A. High order explicit methods for parabolic 
c                equations. BIT, V38,No2,pp.372-390,1998 
c
c
c ------ INPUT PARAMETERS --------------------------
c
c In ordert to use dumka3.f two external procedures are required
c
c double precision FUNCTION RHO (N,T,Y)- returns UPPER estimate of the 
c
c          spectral radius of the Jacobian matrix  of f(t,y) at (t,y). 
c          Must have the following form                 
c                   double precision function rho(n,t,y)
c                   double precision y(n),t    
c                     rho=... 
c                   return
c                   end 
c        for example for y'=-50*y,  you can use any number
c        larger than 50 (for example rho = 51.), or if you have
c        working Euler method with the step size h_{euler}, you can 
c        try to use rho = 2./h_{euler}
c        also N.b. Gerschgorin's theorem can be helpful.
c
c SUBROUTINE FUN(N,T,Y,F) - right hand side of the ODE: y'=f(y,t)
c 
c  N     - number of variables,
c  T     - current time,
C  TEND  - time interval,
c  H0    - initial step size,
c  FUN   - external procedure for right hand evaluation,        
c  Y     - vector of variables,
c     ATOL(*) :   Absolute and relative error tolerances 
c     RTOL(*)     can be both scalar (double precision)
c                 or vectors of length N.
c  IWORK(*):   Integer array of length 2 that gives information 
c                 on how the problem is to be solved
c               
c  IWORK(1):   =0 DUMKA3 attempts to compute the spectral radius 
c                    internally. But you should define a 
c                    dummy function
c
c                    double precision function rho(n,t,y)
c                    double precision y(n),t
c                    integer n
c                    rho = 0
c                    return
c                    end
c
c              =1 RHO is used to obtain upper estimate on 
c                  spectral radius.
c    
c  IWORK(2):   =0 The Jacobian is not constant.                  
c              =1 The Jacobian is constant,
c                 the function rho is called only once
c  IWORK(3):   =0 dumka3 integrates solution until t=tend
c              =1 the code returns after each step t_i chosen
c                    automatically between [t,tend] (solution
c                    at t_i is in y(*) ).
c                    To continue call dumka3 again without changing
c                    any arguments..
c  IWORK(4):   =0 Atol and rtol are scalar (use arrays rtol(1) and atol(1)).
c              =1 Atol and rtol are array of length n (use arrays rtol(n) and atol(n)).
c
c  Z,Z1,Y_EN,Z_EN,OLDEIGENVECTOR  - work arrays size n
c
c ------ OUTPUT PARAMETERS --------------------------
c
c  t         - T-value for which the solution has been computed
c
c  y(n)      - Numerical solution at tend.
c
c
c  IWORK(5)  - Number of function evaluations.
c  IWORK(6)  - Number of steps.
c  IWORK(7)  - Number of accepted steps.
c  IWORK(8)  - Number of rejected steps.
c  IWORK(9)  - Number of evaluations of f used
c                  to estimate the spectral radius
c                  (equal to zero if iwork(1)=1).
c  IWORK(10) - Maximum number of stages used.
c  IWORK(11) - Maximum value of the estimated 
c                  bound for the spectral radius 
c                  (rounded to the nearest integer).
c  IWORK(12) - Minimum value of the estimated  
c                  bound for the spectral radius 
c                  (rounded to the nearest integer).
c
c This program was written by Alexei A. Medovikov.
c
c   E-mail: amedovik@math.tulane.edu
c
c Many thanks Prof. Lebedev V.I. for exellent algorithm of ordering
c roots of stability polynomials, and Prof. Wanner G. and Hairer E.
c for step size control algorithm.
c
c ---------------------------------------------------
      PARAMETER(N_P=16) 
      IMPLICIT DOUBLE PRECISION(A-H,O-Z)
      DOUBLE PRECISION STAB_REG(N_P),COEF(736,6),C_2,C_3,C_4,
     & A_21,A_31,A_41,A_32,A_42,A_43,T,T2,T3,T4,R,COU,H_NEW      
      EXTERNAL FUN
      INTEGER N_DEG(N_P),INDEX_FIRST(N_P),INDEX_LAST(N_P)
      DOUBLE PRECISION Y(*),Z(*),Z1(*),Y_EN(*),Z_EN(*),OLDEIGENVECTOR(*)
      DOUBLE PRECISION  ATOL(*),RTOL(*)
      INTEGER IWORK(12)
      DATA N_DEG/3,6,9,15,21,27,36,48,63,81,135,189,243,324,432,576/
      DATA INDEX_FIRST/1,2,4,7,12,19,28,40,56,77,104,149,212,293,
     &                 401,545/
      DATA INDEX_LAST/1,3,6,11,18,27,39,55,76,103,148,211,292,400,
     &                 544,736/
      DATA STAB_REG/ 2.5005127005D0, 9.784756428464169D0,
     & 23.818760475282560d0,
     & 68.932817124349140D0,136.648186730571300D0,
     & 226.8897061613812D0,404.7232537379578d0,720.7401653373073D0,
     & 1337.1268312643200D0,
     &3266.0271134242240D0,9072.0255133253400D0,17784.544093341440D0,
     &27323.320790121520985D0,50666.52463738415826D0,
     &88860.62470326659968122D0,142602.586088039999594D0/

      DATA (COEF(1,J),J=1,6)/
     &0.924574112262462D0,0.085748762860431D0,0.28771294386877D0,
     &0.085748762860431D0,0.28771294386877D0,0.6265382932708D0/

      DATA ((COEF(I,J),J=1,6),I=2,3)/
c     &.1231817161250036D+00,.1231817161250036D+0,.1042566436020026D+00,
c     &.1231817161250036D+00,.1042566436020026D+0,.3892933924117777D+00,
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c     &.9929505828357067D+00,-.7107751811698341D0,.1010928461953435D+00/
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     &.49334989598763150D0,-0.29116420359694450D0, .18108255547052920D0/


      DATA ((COEF(I,J),J=1,6),I=4,6)/
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       DATA ((COEF(I,J),J=1,6),I=7,11)/
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       DATA ((COEF(I,J),J=1,6),I=12,18)/
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     &.45829477017769808D-3,.87772444441947399D-5,.34880556161053211D-4, 
     &.77848423461079303D-5,.77848423461079303D-5,.70693922388411455D-4, 
     &.77848423461079303D-5,.70693922388411455D-4,.12259411557498746D-4, 
     &.16385315205394168D-4,.16385315205394168D-4,.70492474863571056D-5, 
     &.16385315205394168D-4,.70492474863571056D-5,.13493094613431448D-2, 
     &.92486780710376462D-5,.92486780710376462D-5,.29005112589185273D-4, 
     &.92486780710376462D-5,.29005112589185273D-4,.75552615251588716D-5, 
     &.97635766229904323D-4,.97635766229904323D-4,.11307545519343828D-4, 
     &.97635766229904323D-4,.11307545519343828D-4,.18462543310874277D-4/
 
      DATA ((COEF(I,J),J=1,6),I=705,724)/
     &.71167928983991944D-5,.71167928983991944D-5,.47904006005742867D-3, 
     &.71167928983991944D-5,.47904006005742867D-3,.88013750314859325D-5,
     &.34504615100383551D-4,.34504615100383551D-4,.77708279107186617D-5, 
     &.34504615100383551D-4,.77708279107186617D-5,.71870970162673966D-4, 
     &.12201867415273722D-4,.12201867415273722D-4,.16489249984395503D-4, 
     &.12201867415273722D-4,.16489249984395503D-4,.70520841769397043D-5, 
     &.12528455501298777D-2,.12528455501298777D-2,.98415547727577021D-5, 
     &.12528455501298777D-2,.98415547727577021D-5,.24396027918411155D-4, 
     &.73616190579116149D-5,.73616190579116149D-5,.14791680756823022D-3, 
     &.73616190579116149D-5,.14791680756823022D-3,.10470744926788519D-4, 
     &.21233202323452044D-4,.21233202323452044D-4,.72270639537941898D-5, 
     &.21233202323452044D-4,.72270639537941898D-5,.23632490318042525D-3, 
     &.83802749228177005D-5,.83802749228177005D-5,.42969329942629927D-4, 
     &.83802749228177005D-5,.42969329942629927D-4,.80644521684489196D-5, 
     &.53765588308960295D-4,.53765588308960295D-4,.13404009874506138D-4, 
     &.53765588308960295D-4,.13404009874506138D-4,.14706812328188258D-4, 
     &.70162649105540367D-5,.70162649105540367D-5,.13478768207244246D-1, 
     &.70162649105540367D-5,.13478768207244246D-1,.95162661094599139D-5, 
     &.26654577119062012D-4,.26654577119062012D-4,.74576006200979565D-5, 
     &.26654577119062012D-4,.74576006200979565D-5,.11752463037037148D-3, 
     &.10891824505456188D-4,.10891824505456188D-4,.19689590333047166D-4, 
     &.10891824505456188D-4,.19689590333047166D-4,.71639130531249136D-5, 
     &.33203596934200402D-3,.33203596934200402D-3,.85922871127669669D-5, 
     &.33203596934200402D-3,.85922871127669669D-5,.38143491148269066D-4, 
     &.79025881036570546D-5,.79025881036570546D-5,.62268757844612755D-4, 
     &.79025881036570546D-5,.62268757844612755D-4,.12741699052098109D-4, 
     &.15596299928649304D-4,.15596299928649304D-4,.70303750478911679D-5, 
     &.15596299928649304D-4,.70303750478911679D-5,.27754390877647839D-2, 
     &.10161915328548843D-4,.10161915328548843D-4,.26413113002045082D-4, 
     &.10161915328548843D-4,.26413113002045082D-4,.74474188596923435D-5, 
     &.12011245561186619D-3,.12011245561186619D-3,.10847854073726147D-4, 
     &.12011245561186619D-3,.10847854073726147D-4,.19834929815447473D-4, 
     &.71697114440888450D-5,.71697114440888450D-5,.32003973873601668D-3, 
     &.71697114440888450D-5,.32003973873601668D-3,.85701503562527331D-5, 
     &.38585943320579917D-4,.38585943320579917D-4,.79180255770751269D-5, 
     &.38585943320579917D-4,.79180255770751269D-5,.61326631179228026D-4, 
     &.12804843726408045D-4,.12804843726408045D-4,.15502723882241799D-4, 
     &.12804843726408045D-4,.15502723882241799D-4,.70284915113695867D-5, 
     &.31038237254268014D-2,.31038237254268014D-2,.90291966870403351D-5, 
     &.31038237254268014D-2,.90291966870403351D-5,.31398734883402714D-4/
 

      DATA ((COEF(I,J),J=1,6),I=725,736)/
     &.76514745710831673D-5,.76514745710831673D-5,.83987893343168382D-4, 
     &.76514745710831673D-5,.83987893343168382D-4,.11709437272952442D-4,
     &.17482810818312803D-4,.17482810818312803D-4,.70824371537868323D-5, 
     &.17482810818312803D-4,.70824371537868323D-5,.71128686246649871D-3, 
     &.90029115721813132D-5,.90029115721813132D-5,.31720792790347823D-4, 
     &.90029115721813132D-5,.31720792790347823D-4,.76641446464951208D-5, 
     &.82490990576319028D-4,.82490990576319028D-4,.11761968965338093D-4, 
     &.82490990576319028D-4,.11761968965338093D-4,.17367002047293309D-4, 
     &.70786335148686870D-5,.70786335148686870D-5,.75186114338431098D-3, 
     &.70786335148686870D-5,.75186114338431098D-3,.10199237216518730D-4, 
     &.22627713278614384D-4,.22627713278614384D-4,.72860182531795594D-5, 
     &.22627713278614384D-4,.72860182531795594D-5,.18687871989425724D-3, 
     &.82437233965993013D-5,.82437233965993013D-5,.47575220605350039D-4, 
     &.82437233965993013D-5,.47575220605350039D-4,.13911406857838684D-4, 
     &.10124948250071571D-4,.10124948250071571D-4,.22813183119737957D-4, 
     &.10124948250071571D-4,.22813183119737957D-4,.72782302873419314D-5, 
     &.17697022092294363D-3,.17697022092294363D-3,.70126128917403761D-5, 
     &.17697022092294363D-3,.70126128917403761D-5,.22444828609212017D-4, 
     &.14140914363232827D-4,.14140914363232827D-4,.81880385993792202D-5, 
     &.14140914363232827D-4,.81880385993792202D-5,.46957547824856738D-4, 
     &.13987076721131579D-4,.13987076721131579D-4,.70125085855924287D-5, 
     &.13987076721131579D-4,.70125085855924287D-5,.95474273078995595D-5, 
     &-.47318385459766968D0,.49616903185419430D0,-.25938664641601587D0, 
     &.49616903185419430D0,-.25938664641601587D0,.71215820993688990D-5/

C ---- INPUT VALIDATION
      ZEROVALUE=1.D-14
      IF(H0.LT.ZEROVALUE) THEN
        WRITE(*,*) "INITIAL STEP (H0) IS TOO SMALL"
        RETURN
      ENDIF
      IF (TEND.LE.T) THEN
        WRITE(*,*) "END-TIME (TEND) IS LESS THAN INITIAL TIME (T0)"
        RETURN
      END IF
      IF (IWORK(4).EQ.0) THEN
        IF (ATOL(1).LE.ZEROVALUE.OR.RTOL(1).LE.10.D0*ZEROVALUE) THEN
          WRITE(6,*) 'TOLERANCES ARE TOO SMALL', ATOL(1)," ", RTOL(1)
          RETURN
        END IF
      ELSE
        DO I=1,N
          IF (ATOL(I).LE.ZEROVALUE.OR.RTOL(I).LE.10.D0*ZEROVALUE) THEN
            WRITE(6,*) 'TOLERANCES ARE TOO SMALL'
            RETURN
          END IF
        END DO
      END IF

      IF ( IWORK(1).EQ.0 ) THEN
        WRITE(*,*) 'IWORK(1)=0 - Compute the spectral radius internally'
      ELSE IF (IWORK(1).EQ.1) THEN
        WRITE(*,*) 'IWORK(1)=1 - RHO is used to obtain spectral radius'
      ELSE
         WRITE(*,*) 'INCORRECT IWORK(1) PARAMETER - IT CAN BE 0 OR 1'
      END IF

      IF ( IWORK(2).EQ.0 ) THEN
        WRITE(*,*) 'IWORK(2)=0 - The Jacobian is not constant'
      ELSE IF (IWORK(2).EQ.1) THEN
        WRITE(*,*) 'IWORK(2)=1 - The Jacobian is constant'
      ELSE
         WRITE(*,*) 'INCORRECT IWORK(2) PARAMETER - IT CAN BE 0 OR 1'
      END IF

      IF ( IWORK(3).EQ.0 ) THEN
        WRITE(*,*) 'IWORK(3)=0 - DUMKA3 compute until t=tend'
      ELSE IF (IWORK(3).EQ.1) THEN
        WRITE(*,*) 'IWORK(3)=1 - DUMKA3 returns after each step'
      ELSE
         WRITE(*,*) 'INCORRECT IWORK(3) PARAMETER - IT CAN BE 0 OR 1'
      END IF

      IF ( IWORK(4).EQ.0 ) THEN
        WRITE(*,*) 'IWORK(4)=0 - Atol and rtol are scalar'
      ELSE IF (IWORK(4).EQ.1) THEN
        WRITE(*,*) 'IWORK(4)=1 - Atol and rtol are arrays of length n'
      ELSE
         WRITE(*,*) 'INCORRECT IWORK(4) PARAMETER - IT CAN BE 0 OR 1'
      END IF
C ---- INITIALIZATION -------------------
      IWORK(5)=0
      IWORK(6)=0
      IWORK(7)=0
      IWORK(8)=0
      IWORK(9)=0
      IWORK(10)=0      
      IWORK(11)=0
      IWORK(12)=0
      I_ERR=0      
      ERR_N = 0.D0
      H_N = 0.D0
      H_NEW=H0
      CALL FUN(N,T,Y,Z)
      IWORK(5)=IWORK(5)+1
      IF ( IWORK(1).EQ.0 ) THEN
        CALL FUN(N,T,Z,OLDEIGENVECTOR)
        IWORK(9)=IWORK(9)+1
      ENDIF

200   CONTINUE
C ---- CALCULATE 2/MAX_EIGENVALUE (WHICH IS MAXIMUM STEP 
C ---- SIZE IN EULER METHOD) 
C ---- VIA INTERNAL(EIGENVALUE) OR EXTERNAL(RHO) PROCEDURE      
      IWORK(6)=IWORK(6)+1
      IF ( IWORK(2).EQ.0.OR.IWORK(6).EQ.1) THEN
        IF ( IWORK(1).EQ.0 ) THEN
          IF ((IWORK(6).EQ.1).OR.
     &           (MOD(IWORK(6),20).EQ.0).OR.(I_ERR.EQ.1)) THEN
            EV = DABS( EIGENVALUE(N, Y, Z, OLDEIGENVECTOR, Z1,
     &            EIGENVALUEOLD, T, FUN, ZEROVALUE, IWORK(9)) )
            IF ( EV.LT.ZEROVALUE ) THEN
              EV = ZEROVALUE
            END IF
            COU = 2.D0/ (EV*1.2D0)
           END IF
        ELSE         
          EV = DABS ( RHO(N,T,Y) )
          IF (EV.EQ.0.D0) THEN
            EV = ZEROVALUE
           WRITE(*,*) "RHO RETURNS 0: ", EV," WILL BE USED"
          END IF
          COU = 2.D0/EV
        ENDIF
      ENDIF
      IF ( IWORK(11).LT.EV ) IWORK(11)=EV
      IF ( IWORK(12).GT.EV.OR.IWORK(6).EQ.1 ) IWORK(12)=EV
      
      I_ERR=0
      H=DMIN1(H_NEW,(TEND-T))
      INDEX=1
5     IF((2.D0*H/COU).GT.STAB_REG(INDEX)) THEN
       IF(INDEX.EQ.N_P) GOTO 6
       INDEX=INDEX+1
       GOTO 5
      END IF
6     CONTINUE
      IF(INDEX.GT.1) THEN
      IF(((STAB_REG(INDEX-1)/(N_DEG(INDEX-1)*1.D0))*N_DEG(INDEX)*1.D0)
     &                   .GT.(2.d0*H/COU))THEN
      INDEX=INDEX-1
      END IF
      END IF
      N_POL_DEGREE=N_DEG(INDEX)
      IF(IWORK(10).LT.N_POL_DEGREE) IWORK(10)=N_POL_DEGREE

      H=DMIN1((STAB_REG(INDEX)*COU*0.5d0),H)
C      WRITE(*,300)  N_POL_DEGREE,H/(N_POL_DEGREE*COU),T
300   FORMAT (1X,'SPEED UP=',I3,' TIMES',' H/COU=',F14.7,'  TIME',F14.7)
      N_POL=0


      DO K=INDEX_FIRST(INDEX),INDEX_LAST(INDEX)
      IF(K.EQ.INDEX_FIRST(INDEX)) THEN
               DO I=1,N
               Y_EN(I)=Y(I)
               END DO
               DO I=1,N
               Z_EN(I)=Z(I)
               END DO
               T_EN=T
      END IF
      N_POL=N_POL+3
      A_21=H*COEF(K,1)
      C_2=A_21
      A_31=H*COEF(K,2)
      A_32=H*COEF(K,3)
      C_3=A_31+A_32
      A_41=H*COEF(K,4)
      A_42=H*COEF(K,5)
      A_43=H*COEF(K,6)
      C_4=A_41+A_42+A_43
      DO I=1,N
         Y(I)=Y(I)+A_21*Z(I)
      END DO
      T2=T+C_2
      CALL FUN(N,T2,Y,Z1)
      IWORK(5)=IWORK(5)+1
      IF(N_POL.EQ.N_DEG(INDEX)) THEN
        R=H*(COEF(K,2)-COEF(K,1))
        DO I=1,N
           Y(I)=Y(I)+R*Z(I)+A_32*Z1(I)
        END DO
      ELSE
        DO I=1,N
           Y(I)=Y(I)+A_32*Z1(I)
        END DO
      END IF
      T3=T+C_3
      IF(N_POL.EQ.N_DEG(INDEX)) THEN
	      TMP_1=C_3/2.D0
	      TMP_2=(C_4-C_2)/2.D0
         DO I=1,N
            Z1(I)=TMP_1*Z1(I)-TMP_2*Z(I)
         END DO
      END IF
      CALL FUN(N,T3,Y,Z)
      IWORK(5)=IWORK(5)+1
      IF(N_POL.EQ.N_DEG(INDEX)) THEN
         DO I=1,N
            Z1(I)=TMP_2*Z(I)+Z1(I)
         END DO
      END IF
      DO I=1,N
         Y(I)=Y(I)+A_43*Z(I)
      END DO
      T4=T+C_4
      T=T4
      CALL FUN(N,T,Y,Z)
      IWORK(5)=IWORK(5)+1
      END DO

      IF(IWORK(4).EQ.0) THEN
        ATOL_SCALAR = ATOL(1)
        RTOL_SCALAR = RTOL(1)
        DO I=1,N
          Z1(I)=((-TMP_1)*Z(I)+Z1(I))/
     &        (RTOL_SCALAR*DMAX1(DABS(Y(I)),DABS(Y_EN(I)))+ATOL_SCALAR)       
        END DO
      ELSE
        DO I=1,N
          Z1(I)=((-TMP_1)*Z(I)+Z1(I))/
     &        (RTOL(I)*DMAX1(DABS(Y(I)),DABS(Y_EN(I)))+ATOL(I))       
        END DO
      END IF
C      IF(N_POL_DEGREE.EQ.3) THEN
C        ERROR_CORRECTOR = -3.4785D0/ 8.3338D0
C        DO I=1,N
C          Z1(I)=ERROR_CORRECTOR*Z1(I)
C        END DO
C      END IF        
      H_NEW=H
      CALL ST(N,T,Y,Z1,H_NEW,I_ERR,
     &        N_POL_DEGREE,ERR_N,H_N,IWORK(6))
      IF(I_ERR.EQ.1) THEN         
         DO I=1,N
           Y(I)=Y_EN(I)
         END DO
         DO I=1,N
           Z(I)=Z_EN(I)
         END DO
         T=T_EN
         IWORK(8)=IWORK(8)+1
         GOTO 200
      ELSE
        IWORK(7)=IWORK(7)+1
        IF(T.GE.(TEND).OR.IWORK(3).EQ.1) THEN
          H0=H_NEW
          RETURN          
        ELSE
          GOTO 200
        END IF
      END IF

      RETURN
      END
c
c
c
      SUBROUTINE ST(N,T,Y,Z1,H_NEW,I_ERR,
     &              N_POL_DEGREE,ERR_N,H_N,ISTEP_ID)
      IMPLICIT DOUBLE PRECISION(A-H,O-Z)
      DOUBLE PRECISION Y(N),Z1(N)
      EPS1=0.D0       
      DO I=1,N       
        EPS1=EPS1+Z1(I)*Z1(I)
      END DO    
      EPS=DSQRT( EPS1/(1.D0*N) )
      FRACMIN=0.1D0
      IF(EPS.EQ.0.D0) EPS=1.D-14       
      FRAC=(1.D0/EPS)**(1.D0/3.D0)
      IF(EPS.LE.1.D0) THEN
        IF (ERR_N.GT.0.D0.AND.H_N.GT.0.D0) THEN
          FRAC2=ERR_N**(1.D0/3.D0)*FRAC**2.d0*(H_NEW/H_N)
          FRAC=DMIN1(FRAC,FRAC2)
        END IF
        I_ERR=0 
        FRACMAX=2.D0
        FRAC=DMIN1(FRACMAX,DMAX1(FRACMIN,0.8D0*FRAC))
        H_OLD = H_NEW
        H_NEW=FRAC*H_NEW            
        H_N=H_OLD 
        ERR_N=EPS 
C        WRITE(*,102) EPS, T
C        WRITE(*,101) N_POL_DEGREE, H_NEW, H_OLD      
      ELSE
        I_ERR=1
        FRACMAX=1.D0
        FRAC=0.8D0*DMIN1(FRACMAX,DMAX1(FRACMIN,0.8D0*FRAC))
        IF(ISTEP_ID.EQ.1) THEN
          H_NEW=FRACMIN*H_NEW
        ELSE
          H_NEW=FRAC*H_NEW
        ENDIF
        WRITE(*,100) EPS, T
        WRITE(*,101) N_POL_DEGREE, H_NEW, H_OLD 
      END IF

100   FORMAT('STEP REJECTED, BECAUSE ERROR=',1E18.10,
     &       ' TIME=',1E18.10)
102   FORMAT('STEP ACCEPTED, BECAUSE ERROR=',1E18.10,
     &       ' TIME=',1E18.10)
101   FORMAT('  POLYNOMIAL DEGREE =', I3,
     &       ' NEW STEP =',1E18.10,' OLD STEP',1E18.10)
      RETURN
      END

		  DOUBLE PRECISION FUNCTION EIGENVALUE(N,Y,FY,V,
     &             TMP,EIGENVALUEOLD,T,FUN,ZEROVALUE,NUM_FUN_EVAL)
      IMPLICIT DOUBLE PRECISION(A-H,O-Z)
      EXTERNAL FUN
      DOUBLE PRECISION Y(N),FY(N),V(N),TMP(N)
		    WRITE(*,*) "COMPUTE EIGENVALUE ..."
    		MAXIT = 30
    		TOL = 0.01
    		IDX = 0
    		RADIUS = 1.D-10
    		ALMBDA = EIGENVALUEOLD
    		DNRMV = 0.D0  
     		DNRMV = DNORM(N,V)
    		DO WHILE ( IDX.EQ.0.OR.(DABS(ALMBDA - EIGENVALUEOLD)
     &          .GT.DABS(ALMBDA)*TOL.AND.IDX.LE.MAXIT ) )
      		IDX = IDX+1            
C      		ALL VECTORS SHOULD BE ||Y-V|| < RADIUS, BECAUSE PROBLEM CAN BE NON-LINEAR
      		IF ( DNRMV.LT.ZEROVALUE ) THEN
           	DO I=1,N	
             	V(I) = RAND(1)               
           	END DO
            DNRMV = DNORM(N,V)
          ENDIF
       		DNORMINV= RADIUS/DNRMV
       		DO I=1,N
         		V(I) = Y(I) + DNORMINV * V(I)            
       		END DO
          CALL FUN(N,T,V,TMP)
          NUM_FUN_EVAL=NUM_FUN_EVAL+1
C      		APPROXIMATION OF EIGENVECTOR
      		DO I=1,N
        	  V(I) = TMP(I) - FY(I)
      		END DO
C      		// APPROXIMATION OF EIGENVALUE
      		DNRMV = DNORM(N,V)
          EIGENVALUEOLD = ALMBDA
      		ALMBDA = DNRMV/RADIUS
          WRITE(*,*) "ITERATION = ", IDX, " EIGENVALUE = ", ALMBDA    		
    	  END DO
    	  EIGENVALUEOLD = ALMBDA
    	  EIGENVALUE = ALMBDA
        RETURN 
        END

  	    DOUBLE PRECISION FUNCTION DNORM ( N, Y ) 
        IMPLICIT DOUBLE PRECISION(A-H,O-Z)
        DOUBLE PRECISION Y(N)
          DNORM = 0.D0       
          DO I=1,N
      	    DNORM = DNORM + Y(I)*Y(I)
          END DO
    	  DNORM = DSQRT(DNORM)
    	  RETURN
        END
