C     SUBROUTINE NNLS  (A,MDA,M,N,B,X,RNORM,W,ZZ,INDEX,MODE)
C   
C  Algorithm NNLS: NONNEGATIVE LEAST SQUARES
C   
c  The original version of this code was developed by
c  Charles L. Lawson and Richard J. Hanson at Jet Propulsion Laboratory
c  1973 JUN 15, and published in the book
c  "SOLVING LEAST SQUARES PROBLEMS", Prentice-HalL, 1974.
c  Revised FEB 1995 to accompany reprinting of the book by SIAM.
c
C     GIVEN AN M BY N MATRIX, A, AND AN M-VECTOR, B,  COMPUTE AN
C     N-VECTOR, X, THAT SOLVES THE LEAST SQUARES PROBLEM   
C   
C                      A * X = B  SUBJECT TO X .GE. 0   
C     ------------------------------------------------------------------
c                     Subroutine Arguments
c
C     A(),MDA,M,N     MDA IS THE FIRST DIMENSIONING PARAMETER FOR THE   
C                     ARRAY, A().   ON ENTRY A() CONTAINS THE M BY N    
C                     MATRIX, A.           ON EXIT A() CONTAINS 
C                     THE PRODUCT MATRIX, Q*A , WHERE Q IS AN   
C                     M BY M ORTHOGONAL MATRIX GENERATED IMPLICITLY BY  
C                     THIS SUBROUTINE.  
C     B()     ON ENTRY B() CONTAINS THE M-VECTOR, B.   ON EXIT B() CON- 
C             TAINS Q*B.
C     X()     ON ENTRY X() NEED NOT BE INITIALIZED.  ON EXIT X() WILL   
C             CONTAIN THE SOLUTION VECTOR.  
C     RNORM   ON EXIT RNORM CONTAINS THE EUCLIDEAN NORM OF THE  
C             RESIDUAL VECTOR.  
C     W()     AN N-ARRAY OF WORKING SPACE.  ON EXIT W() WILL CONTAIN    
C             THE DUAL SOLUTION VECTOR.   W WILL SATISFY W(I) = 0.  
C             FOR ALL I IN SET P  AND W(I) .LE. 0. FOR ALL I IN SET Z   
C     ZZ()     AN M-ARRAY OF WORKING SPACE.     
C     INDEX()     AN INTEGER WORKING ARRAY OF LENGTH AT LEAST N.
C                 ON EXIT THE CONTENTS OF THIS ARRAY DEFINE THE SETS    
C                 P AND Z AS FOLLOWS..  
C   
C                 INDEX(1)   THRU INDEX(NSETP) = SET P.     
C                 INDEX(IZ1) THRU INDEX(IZ2)   = SET Z.     
C                 IZ1 = NSETP + 1 = NPP1
C                 IZ2 = N   
C     MODE    THIS IS A SUCCESS-FAILURE FLAG WITH THE FOLLOWING 
C             MEANINGS. 
C             1     THE SOLUTION HAS BEEN COMPUTED SUCCESSFULLY.
C             2     THE DIMENSIONS OF THE PROBLEM ARE BAD.  
C                   EITHER M .LE. 0 OR N .LE. 0.
C             3    ITERATION COUNT EXCEEDED.  MORE THAN 3*N ITERATIONS. 
C   
C     ------------------------------------------------------------------
      SUBROUTINE NNLS (A,MDA,M,N,B,X,RNORM,W,ZZ,INDEX,MODE) 
C     ------------------------------------------------------------------
      integer I, II, IP, ITER, ITMAX, IZ, IZ1, IZ2, IZMAX, J, JJ, JZ, L
      integer M, MDA, MODE,N, NPP1, NSETP, RTNKEY
c     integer INDEX(N)  
c     double precision A(MDA,N), B(M), W(N), X(N), ZZ(M)   
      integer INDEX(*)  
      double precision A(MDA,*), B(*), W(*), X(*), ZZ(*)   
      double precision ALPHA, ASAVE, CC, DIFF, DUMMY, FACTOR, RNORM
      double precision SM, SS, T, TEMP, TWO, UNORM, UP, WMAX
      double precision ZERO, ZTEST
      parameter(FACTOR = 0.01d0)
      parameter(TWO = 2.0d0, ZERO = 0.0d0)
C     ------------------------------------------------------------------
      MODE=1
      IF (M .le. 0 .or. N .le. 0) then
         MODE=2
         RETURN
      endif
      ITER=0
      ITMAX=3*N 
C   
C                    INITIALIZE THE ARRAYS INDEX() AND X(). 
C   
          DO 20 I=1,N   
          X(I)=ZERO     
   20     INDEX(I)=I    
C   
      IZ2=N 
      IZ1=1 
      NSETP=0   
      NPP1=1
C                             ******  MAIN LOOP BEGINS HERE  ******     
   30 CONTINUE  
C                  QUIT IF ALL COEFFICIENTS ARE ALREADY IN THE SOLUTION.
C                        OR IF M COLS OF A HAVE BEEN TRIANGULARIZED.    
C   
      IF (IZ1 .GT.IZ2.OR.NSETP.GE.M) GO TO 350   
C   
C         COMPUTE COMPONENTS OF THE DUAL (NEGATIVE GRADIENT) VECTOR W().
C   
      DO 50 IZ=IZ1,IZ2  
         J=INDEX(IZ)   
         SM=ZERO   
         DO 40 L=NPP1,M
   40        SM=SM+A(L,J)*B(L)     
         W(J)=SM   
   50 continue
C                                   FIND LARGEST POSITIVE W(J). 
   60 continue
      WMAX=ZERO 
      DO 70 IZ=IZ1,IZ2  
         J=INDEX(IZ)   
         IF (W(J) .gt. WMAX) then
            WMAX=W(J)     
            IZMAX=IZ  
         endif
   70 CONTINUE  
C   
C             IF WMAX .LE. 0. GO TO TERMINATION.
C             THIS INDICATES SATISFACTION OF THE KUHN-TUCKER CONDITIONS.
C   
      IF (WMAX .le. ZERO) go to 350
      IZ=IZMAX  
      J=INDEX(IZ)   
C   
C     THE SIGN OF W(J) IS OK FOR J TO BE MOVED TO SET P.    
C     BEGIN THE TRANSFORMATION AND CHECK NEW DIAGONAL ELEMENT TO AVOID  
C     NEAR LINEAR DEPENDENCE.   
C   
      ASAVE=A(NPP1,J)   
      CALL H12 (1,NPP1,NPP1+1,M,A(1,J),1,UP,DUMMY,1,1,0)    
      UNORM=ZERO
      IF (NSETP .ne. 0) then
          DO 90 L=1,NSETP   
   90       UNORM=UNORM+A(L,J)**2     
      endif
      UNORM=sqrt(UNORM) 
      IF (DIFF(UNORM+ABS(A(NPP1,J))*FACTOR,UNORM) .gt. ZERO) then
C   
C        COL J IS SUFFICIENTLY INDEPENDENT.  COPY B INTO ZZ, UPDATE ZZ
C        AND SOLVE FOR ZTEST ( = PROPOSED NEW VALUE FOR X(J) ).    
C   
         DO 120 L=1,M  
  120        ZZ(L)=B(L)    
         CALL H12 (2,NPP1,NPP1+1,M,A(1,J),1,UP,ZZ,1,1,1)   
         ZTEST=ZZ(NPP1)/A(NPP1,J)  
C   
C                                     SEE IF ZTEST IS POSITIVE  
C   
         IF (ZTEST .gt. ZERO) go to 140
      endif
C   
C     REJECT J AS A CANDIDATE TO BE MOVED FROM SET Z TO SET P.  
C     RESTORE A(NPP1,J), SET W(J)=0., AND LOOP BACK TO TEST DUAL
C     COEFFS AGAIN.     
C   
      A(NPP1,J)=ASAVE   
      W(J)=ZERO 
      GO TO 60  
C   
C     THE INDEX  J=INDEX(IZ)  HAS BEEN SELECTED TO BE MOVED FROM
C     SET Z TO SET P.    UPDATE B,  UPDATE INDICES,  APPLY HOUSEHOLDER  
C     TRANSFORMATIONS TO COLS IN NEW SET Z,  ZERO SUBDIAGONAL ELTS IN   
C     COL J,  SET W(J)=0.   
C   
  140 continue
      DO 150 L=1,M  
  150    B(L)=ZZ(L)    
C   
      INDEX(IZ)=INDEX(IZ1)  
      INDEX(IZ1)=J  
      IZ1=IZ1+1 
      NSETP=NPP1
      NPP1=NPP1+1   
C   
      IF (IZ1 .le. IZ2) then
         DO 160 JZ=IZ1,IZ2 
            JJ=INDEX(JZ)  
            CALL H12 (2,NSETP,NPP1,M,A(1,J),1,UP,A(1,JJ),1,MDA,1)
  160    continue
      endif
C   
      IF (NSETP .ne. M) then
         DO 180 L=NPP1,M   
  180       A(L,J)=ZERO   
      endif
C   
      W(J)=ZERO 
C                                SOLVE THE TRIANGULAR SYSTEM.   
C                                STORE THE SOLUTION TEMPORARILY IN ZZ().
      RTNKEY = 1
      GO TO 400 
  200 CONTINUE  
C   
C                       ******  SECONDARY LOOP BEGINS HERE ******   
C   
C                          ITERATION COUNTER.   
C 
  210 continue  
      ITER=ITER+1   
      IF (ITER .gt. ITMAX) then
         MODE=3
         write (*,'(/a)') ' NNLS quitting on iteration count.'
         GO TO 350 
      endif
C   
C                    SEE IF ALL NEW CONSTRAINED COEFFS ARE FEASIBLE.    
C                                  IF NOT COMPUTE ALPHA.    
C   
      ALPHA=TWO 
      DO 240 IP=1,NSETP 
         L=INDEX(IP)   
         IF (ZZ(IP) .le. ZERO) then
            T=-X(L)/(ZZ(IP)-X(L))     
            IF (ALPHA .gt. T) then
               ALPHA=T   
               JJ=IP 
            endif
         endif
  240 CONTINUE  
C   
C          IF ALL NEW CONSTRAINED COEFFS ARE FEASIBLE THEN ALPHA WILL   
C          STILL = 2.    IF SO EXIT FROM SECONDARY LOOP TO MAIN LOOP.   
C   
      IF (ALPHA.EQ.TWO) GO TO 330   
C   
C          OTHERWISE USE ALPHA WHICH WILL BE BETWEEN 0. AND 1. TO   
C          INTERPOLATE BETWEEN THE OLD X AND THE NEW ZZ.    
C   
      DO 250 IP=1,NSETP 
         L=INDEX(IP)   
         X(L)=X(L)+ALPHA*(ZZ(IP)-X(L)) 
  250 continue
C   
C        MODIFY A AND B AND THE INDEX ARRAYS TO MOVE COEFFICIENT I  
C        FROM SET P TO SET Z.   
C   
      I=INDEX(JJ)   
  260 continue
      X(I)=ZERO 
C   
      IF (JJ .ne. NSETP) then
         JJ=JJ+1   
         DO 280 J=JJ,NSETP 
            II=INDEX(J)   
            INDEX(J-1)=II 
            CALL G1 (A(J-1,II),A(J,II),CC,SS,A(J-1,II))   
            A(J,II)=ZERO  
            DO 270 L=1,N  
               IF (L.NE.II) then
c
c                 Apply procedure G2 (CC,SS,A(J-1,L),A(J,L))  
c
                  TEMP = A(J-1,L)
                  A(J-1,L) = CC*TEMP + SS*A(J,L)
                  A(J,L)   =-SS*TEMP + CC*A(J,L)
               endif
  270       CONTINUE  
c
c                 Apply procedure G2 (CC,SS,B(J-1),B(J))   
c
            TEMP = B(J-1)
            B(J-1) = CC*TEMP + SS*B(J)    
            B(J)   =-SS*TEMP + CC*B(J)    
  280    continue
      endif
c
      NPP1=NSETP
      NSETP=NSETP-1     
      IZ1=IZ1-1 
      INDEX(IZ1)=I  
C   
C        SEE IF THE REMAINING COEFFS IN SET P ARE FEASIBLE.  THEY SHOULD
C        BE BECAUSE OF THE WAY ALPHA WAS DETERMINED.
C        IF ANY ARE INFEASIBLE IT IS DUE TO ROUND-OFF ERROR.  ANY   
C        THAT ARE NONPOSITIVE WILL BE SET TO ZERO   
C        AND MOVED FROM SET P TO SET Z. 
C   
      DO 300 JJ=1,NSETP 
         I=INDEX(JJ)   
         IF (X(I) .le. ZERO) go to 260
  300 CONTINUE  
C   
C         COPY B( ) INTO ZZ( ).  THEN SOLVE AGAIN AND LOOP BACK.
C   
      DO 310 I=1,M  
  310     ZZ(I)=B(I)    
      RTNKEY = 2
      GO TO 400 
  320 CONTINUE  
      GO TO 210 
C                      ******  END OF SECONDARY LOOP  ******
C   
  330 continue
      DO 340 IP=1,NSETP 
          I=INDEX(IP)   
  340     X(I)=ZZ(IP)   
C        ALL NEW COEFFS ARE POSITIVE.  LOOP BACK TO BEGINNING.  
      GO TO 30  
C   
C                        ******  END OF MAIN LOOP  ******   
C   
C                        COME TO HERE FOR TERMINATION.  
C                     COMPUTE THE NORM OF THE FINAL RESIDUAL VECTOR.    
C 
  350 continue  
      SM=ZERO   
      IF (NPP1 .le. M) then
         DO 360 I=NPP1,M   
  360       SM=SM+B(I)**2 
      else
         DO 380 J=1,N  
  380       W(J)=ZERO     
      endif
      RNORM=sqrt(SM)    
      RETURN
C   
C     THE FOLLOWING BLOCK OF CODE IS USED AS AN INTERNAL SUBROUTINE     
C     TO SOLVE THE TRIANGULAR SYSTEM, PUTTING THE SOLUTION IN ZZ().     
C   
  400 continue
      DO 430 L=1,NSETP  
         IP=NSETP+1-L  
         IF (L .ne. 1) then
            DO 410 II=1,IP
               ZZ(II)=ZZ(II)-A(II,JJ)*ZZ(IP+1)   
  410       continue
         endif
         JJ=INDEX(IP)  
         ZZ(IP)=ZZ(IP)/A(IP,JJ)    
  430 continue
      go to (200, 320), RTNKEY
      END   


       double precision FUNCTION DIFF(X,Y)
c
c  Function used in tests that depend on machine precision.
c
c  The original version of this code was developed by
c  Charles L. Lawson and Richard J. Hanson at Jet Propulsion Laboratory
c  1973 JUN 7, and published in the book
c  "SOLVING LEAST SQUARES PROBLEMS", Prentice-HalL, 1974.
c  Revised FEB 1995 to accompany reprinting of the book by SIAM.
C
      double precision X, Y
      DIFF=X-Y  
      RETURN
      END   


C     SUBROUTINE H12 (MODE,LPIVOT,L1,M,U,IUE,UP,C,ICE,ICV,NCV)  
C   
C  CONSTRUCTION AND/OR APPLICATION OF A SINGLE   
C  HOUSEHOLDER TRANSFORMATION..     Q = I + U*(U**T)/B   
C   
c  The original version of this code was developed by
c  Charles L. Lawson and Richard J. Hanson at Jet Propulsion Laboratory
c  1973 JUN 12, and published in the book
c  "SOLVING LEAST SQUARES PROBLEMS", Prentice-HalL, 1974.
c  Revised FEB 1995 to accompany reprinting of the book by SIAM.
C     ------------------------------------------------------------------
c                     Subroutine Arguments
c
C     MODE   = 1 OR 2   Selects Algorithm H1 to construct and apply a
c            Householder transformation, or Algorithm H2 to apply a
c            previously constructed transformation.
C     LPIVOT IS THE INDEX OF THE PIVOT ELEMENT. 
C     L1,M   IF L1 .LE. M   THE TRANSFORMATION WILL BE CONSTRUCTED TO   
C            ZERO ELEMENTS INDEXED FROM L1 THROUGH M.   IF L1 GT. M     
C            THE SUBROUTINE DOES AN IDENTITY TRANSFORMATION.
C     U(),IUE,UP    On entry with MODE = 1, U() contains the pivot
c            vector.  IUE is the storage increment between elements.  
c            On exit when MODE = 1, U() and UP contain quantities
c            defining the vector U of the Householder transformation.
c            on entry with MODE = 2, U() and UP should contain
c            quantities previously computed with MODE = 1.  These will
c            not be modified during the entry with MODE = 2.   
C     C()    ON ENTRY with MODE = 1 or 2, C() CONTAINS A MATRIX WHICH
c            WILL BE REGARDED AS A SET OF VECTORS TO WHICH THE
c            HOUSEHOLDER TRANSFORMATION IS TO BE APPLIED.
c            ON EXIT C() CONTAINS THE SET OF TRANSFORMED VECTORS.
C     ICE    STORAGE INCREMENT BETWEEN ELEMENTS OF VECTORS IN C().  
C     ICV    STORAGE INCREMENT BETWEEN VECTORS IN C().  
C     NCV    NUMBER OF VECTORS IN C() TO BE TRANSFORMED. IF NCV .LE. 0  
C            NO OPERATIONS WILL BE DONE ON C(). 
C     ------------------------------------------------------------------
      SUBROUTINE H12 (MODE,LPIVOT,L1,M,U,IUE,UP,C,ICE,ICV,NCV)  
C     ------------------------------------------------------------------
      integer I, I2, I3, I4, ICE, ICV, INCR, IUE, J
      integer L1, LPIVOT, M, MODE, NCV
      double precision B, C(*), CL, CLINV, ONE, SM
c     double precision U(IUE,M)
      double precision U(IUE,*)
      double precision UP
      parameter(ONE = 1.0d0)
C     ------------------------------------------------------------------
      IF (0.GE.LPIVOT.OR.LPIVOT.GE.L1.OR.L1.GT.M) RETURN    
      CL=abs(U(1,LPIVOT))   
      IF (MODE.EQ.2) GO TO 60   
C                            ****** CONSTRUCT THE TRANSFORMATION. ******
          DO 10 J=L1,M  
   10     CL=MAX(abs(U(1,J)),CL)  
      IF (CL) 130,130,20
   20 CLINV=ONE/CL  
      SM=(U(1,LPIVOT)*CLINV)**2   
          DO 30 J=L1,M  
   30     SM=SM+(U(1,J)*CLINV)**2 
      CL=CL*SQRT(SM)   
      IF (U(1,LPIVOT)) 50,50,40     
   40 CL=-CL
   50 UP=U(1,LPIVOT)-CL 
      U(1,LPIVOT)=CL    
      GO TO 70  
C            ****** APPLY THE TRANSFORMATION  I+U*(U**T)/B  TO C. ******
C   
   60 IF (CL) 130,130,70
   70 IF (NCV.LE.0) RETURN  
      B= UP*U(1,LPIVOT)
C                       B  MUST BE NONPOSITIVE HERE.  IF B = 0., RETURN.
C   
      IF (B) 80,130,130 
   80 B=ONE/B   
      I2=1-ICV+ICE*(LPIVOT-1)   
      INCR=ICE*(L1-LPIVOT)  
          DO 120 J=1,NCV
          I2=I2+ICV     
          I3=I2+INCR    
          I4=I3 
          SM=C(I2)*UP
              DO 90 I=L1,M  
              SM=SM+C(I3)*U(1,I)
   90         I3=I3+ICE 
          IF (SM) 100,120,100   
  100     SM=SM*B   
          C(I2)=C(I2)+SM*UP
              DO 110 I=L1,M 
              C(I4)=C(I4)+SM*U(1,I)
  110         I4=I4+ICE 
  120     CONTINUE  
  130 RETURN
      END   



      SUBROUTINE G1 (A,B,CTERM,STERM,SIG)   
c
C     COMPUTE ORTHOGONAL ROTATION MATRIX..  
c
c  The original version of this code was developed by
c  Charles L. Lawson and Richard J. Hanson at Jet Propulsion Laboratory
c  1973 JUN 12, and published in the book
c  "SOLVING LEAST SQUARES PROBLEMS", Prentice-HalL, 1974.
c  Revised FEB 1995 to accompany reprinting of the book by SIAM.
C   
C     COMPUTE.. MATRIX   (C, S) SO THAT (C, S)(A) = (SQRT(A**2+B**2))   
C                        (-S,C)         (-S,C)(B)   (   0          )    
C     COMPUTE SIG = SQRT(A**2+B**2) 
C        SIG IS COMPUTED LAST TO ALLOW FOR THE POSSIBILITY THAT 
C        SIG MAY BE IN THE SAME LOCATION AS A OR B .
C     ------------------------------------------------------------------
      double precision A, B, CTERM, ONE, SIG, STERM, XR, YR, ZERO
      parameter(ONE = 1.0d0, ZERO = 0.0d0)
C     ------------------------------------------------------------------
      if (abs(A) .gt. abs(B)) then
         XR=B/A
         YR=sqrt(ONE+XR**2)
         CTERM=sign(ONE/YR,A)
         STERM=CTERM*XR
         SIG=abs(A)*YR     
         RETURN
      endif

      if (B .ne. ZERO) then
         XR=A/B
         YR=sqrt(ONE+XR**2)
         STERM=sign(ONE/YR,B)
         CTERM=STERM*XR
         SIG=abs(B)*YR     
         RETURN
      endif

      SIG=ZERO  
      CTERM=ZERO  
      STERM=ONE   
      RETURN
      END   
