*DECK PCOEF SUBROUTINE PCOEF (L, C, TC, A) C***BEGIN PROLOGUE PCOEF C***PURPOSE Convert the POLFIT coefficients to Taylor series form. C***LIBRARY SLATEC C***CATEGORY K1A1A2 C***TYPE SINGLE PRECISION (PCOEF-S, DPCOEF-D) C***KEYWORDS CURVE FITTING, DATA FITTING, LEAST SQUARES, POLYNOMIAL FIT C***AUTHOR Shampine, L. F., (SNLA) C Davenport, S. M., (SNLA) C***DESCRIPTION C C Written BY L. F. Shampine and S. M. Davenport. C C Abstract C C POLFIT computes the least squares polynomial fit of degree L as C a sum of orthogonal polynomials. PCOEF changes this fit to its C Taylor expansion about any point C , i.e. writes the polynomial C as a sum of powers of (X-C). Taking C=0. gives the polynomial C in powers of X, but a suitable non-zero C often leads to C polynomials which are better scaled and more accurately evaluated. C C The parameters for PCOEF are C C INPUT -- C L - Indicates the degree of polynomial to be changed to C its Taylor expansion. To obtain the Taylor C coefficients in reverse order, input L as the C negative of the degree desired. The absolute value C of L must be less than or equal to NDEG, the highest C degree polynomial fitted by POLFIT . C C - The point about which the Taylor expansion is to be C made. C A - Work and output array containing values from last C call to POLFIT . C C OUTPUT -- C TC - Vector containing the first LL+1 Taylor coefficients C where LL=ABS(L). If L.GT.0 , the coefficients are C in the usual Taylor series order, i.e. C P(X) = TC(1) + TC(2)*(X-C) + ... + TC(N+1)*(X-C)**N C If L .LT. 0, the coefficients are in reverse order, C i.e. C P(X) = TC(1)*(X-C)**N + ... + TC(N)*(X-C) + TC(N+1) C C***REFERENCES L. F. Shampine, S. M. Davenport and R. E. Huddleston, C Curve fitting by polynomials in one variable, Report C SLA-74-0270, Sandia Laboratories, June 1974. C***ROUTINES CALLED PVALUE C***REVISION HISTORY (YYMMDD) C 740601 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE PCOEF C DIMENSION A(*), TC(*) C***FIRST EXECUTABLE STATEMENT PCOEF LL = ABS(L) LLP1 = LL + 1 CALL PVALUE (LL,LL,C,TC(1),TC(2),A) IF (LL .LT. 2) GO TO 2 FAC = 1.0 DO 1 I = 3,LLP1 FAC = FAC*(I-1) 1 TC(I) = TC(I)/FAC 2 IF (L .GE. 0) GO TO 4 NR = LLP1/2 LLP2 = LL + 2 DO 3 I = 1,NR SAVE = TC(I) NEW = LLP2 - I TC(I) = TC(NEW) 3 TC(NEW) = SAVE 4 RETURN END c$$$ c$$$ subroutine dscal(n,da,dx,incx) c$$$c c$$$c scales a vector by a constant. c$$$c uses unrolled loops for increment equal to one. c$$$c jack dongarra, linpack, 3/11/78. c$$$c modified 3/93 to return if incx .le. 0. c$$$c c$$$ double precision da,dx(1) c$$$ integer i,incx,m,mp1,n,nincx c$$$c c$$$ if( n.le.0 .or. incx.le.0 )return c$$$ if(incx.eq.1)go to 20 c$$$c c$$$c code for increment not equal to 1 c$$$c c$$$ nincx = n*incx c$$$ do 10 i = 1,nincx,incx c$$$ dx(i) = da*dx(i) c$$$ 10 continue c$$$ return c$$$c c$$$c code for increment equal to 1 c$$$c c$$$c c$$$c clean-up loop c$$$c c$$$ 20 m = mod(n,5) c$$$ if( m .eq. 0 ) go to 40 c$$$ do 30 i = 1,m c$$$ dx(i) = da*dx(i) c$$$ 30 continue c$$$ if( n .lt. 5 ) return c$$$ 40 mp1 = m + 1 c$$$ do 50 i = mp1,n,5 c$$$ dx(i) = da*dx(i) c$$$ dx(i + 1) = da*dx(i + 1) c$$$ dx(i + 2) = da*dx(i + 2) c$$$ dx(i + 3) = da*dx(i + 3) c$$$ dx(i + 4) = da*dx(i + 4) c$$$ 50 continue c$$$ return c$$$ end subroutine dgbco(abd,lda,n,ml,mu,ipvt,rcond,z) integer lda,n,ml,mu,ipvt(1) double precision abd(lda,1),z(1) double precision rcond c c dgbco factors a double precision band matrix by gaussian c elimination and estimates the condition of the matrix. c c if rcond is not needed, dgbfa is slightly faster. c to solve a*x = b , follow dgbco by dgbsl. c to compute inverse(a)*c , follow dgbco by dgbsl. c to compute determinant(a) , follow dgbco by dgbdi. c c on entry c c abd double precision(lda, n) c contains the matrix in band storage. the columns c of the matrix are stored in the columns of abd and c the diagonals of the matrix are stored in rows c ml+1 through 2*ml+mu+1 of abd . c see the comments below for details. c c lda integer c the leading dimension of the array abd . c lda must be .ge. 2*ml + mu + 1 . c c n integer c the order of the original matrix. c c ml integer c number of diagonals below the main diagonal. c 0 .le. ml .lt. n . c c mu integer c number of diagonals above the main diagonal. c 0 .le. mu .lt. n . c more efficient if ml .le. mu . c c on return c c abd an upper triangular matrix in band storage and c the multipliers which were used to obtain it. c the factorization can be written a = l*u where c l is a product of permutation and unit lower c triangular matrices and u is upper triangular. c c ipvt integer(n) c an integer vector of pivot indices. c c rcond double precision c an estimate of the reciprocal condition of a . c for the system a*x = b , relative perturbations c in a and b of size epsilon may cause c relative perturbations in x of size epsilon/rcond . c if rcond is so small that the logical expression c 1.0 + rcond .eq. 1.0 c is true, then a may be singular to working c precision. in particular, rcond is zero if c exact singularity is detected or the estimate c underflows. c c z double precision(n) c a work vector whose contents are usually unimportant. c if a is close to a singular matrix, then z is c an approximate null vector in the sense that c norm(a*z) = rcond*norm(a)*norm(z) . c c band storage c c if a is a band matrix, the following program segment c will set up the input. c c ml = (band width below the diagonal) c mu = (band width above the diagonal) c m = ml + mu + 1 c do 20 j = 1, n c i1 = max0(1, j-mu) c i2 = min0(n, j+ml) c do 10 i = i1, i2 c k = i - j + m c abd(k,j) = a(i,j) c 10 continue c 20 continue c c this uses rows ml+1 through 2*ml+mu+1 of abd . c in addition, the first ml rows in abd are used for c elements generated during the triangularization. c the total number of rows needed in abd is 2*ml+mu+1 . c the ml+mu by ml+mu upper left triangle and the c ml by ml lower right triangle are not referenced. c c example.. if the original matrix is c c 11 12 13 0 0 0 c 21 22 23 24 0 0 c 0 32 33 34 35 0 c 0 0 43 44 45 46 c 0 0 0 54 55 56 c 0 0 0 0 65 66 c c then n = 6, ml = 1, mu = 2, lda .ge. 5 and abd should contain c c * * * + + + , * = not used c * * 13 24 35 46 , + = used for pivoting c * 12 23 34 45 56 c 11 22 33 44 55 66 c 21 32 43 54 65 * c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c linpack dgbfa c blas daxpy,ddot,dscal,dasum c fortran dabs,dmax1,max0,min0,dsign c c internal variables c double precision ddot,ek,t,wk,wkm double precision anorm,s,dasum,sm,ynorm integer is,info,j,ju,k,kb,kp1,l,la,lm,lz,m,mm c c c compute 1-norm of a c anorm = 0.0d0 l = ml + 1 is = l + mu do 10 j = 1, n anorm = dmax1(anorm,dasum(l,abd(is,j),1)) if (is .gt. ml + 1) is = is - 1 if (j .le. mu) l = l + 1 if (j .ge. n - ml) l = l - 1 10 continue c c factor c call dgbfa(abd,lda,n,ml,mu,ipvt,info) c c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) . c estimate = norm(z)/norm(y) where a*z = y and trans(a)*y = e . c trans(a) is the transpose of a . the components of e are c chosen to cause maximum local growth in the elements of w where c trans(u)*w = e . the vectors are frequently rescaled to avoid c overflow. c c solve trans(u)*w = e c ek = 1.0d0 do 20 j = 1, n z(j) = 0.0d0 20 continue m = ml + mu + 1 ju = 0 do 100 k = 1, n if (z(k) .ne. 0.0d0) ek = dsign(ek,-z(k)) if (dabs(ek-z(k)) .le. dabs(abd(m,k))) go to 30 s = dabs(abd(m,k))/dabs(ek-z(k)) call dscal(n,s,z,1) ek = s*ek 30 continue wk = ek - z(k) wkm = -ek - z(k) s = dabs(wk) sm = dabs(wkm) if (abd(m,k) .eq. 0.0d0) go to 40 wk = wk/abd(m,k) wkm = wkm/abd(m,k) go to 50 40 continue wk = 1.0d0 wkm = 1.0d0 50 continue kp1 = k + 1 ju = min0(max0(ju,mu+ipvt(k)),n) mm = m if (kp1 .gt. ju) go to 90 do 60 j = kp1, ju mm = mm - 1 sm = sm + dabs(z(j)+wkm*abd(mm,j)) z(j) = z(j) + wk*abd(mm,j) s = s + dabs(z(j)) 60 continue if (s .ge. sm) go to 80 t = wkm - wk wk = wkm mm = m do 70 j = kp1, ju mm = mm - 1 z(j) = z(j) + t*abd(mm,j) 70 continue 80 continue 90 continue z(k) = wk 100 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c c solve trans(l)*y = w c do 120 kb = 1, n k = n + 1 - kb lm = min0(ml,n-k) if (k .lt. n) z(k) = z(k) + ddot(lm,abd(m+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 110 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) 110 continue l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t 120 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c ynorm = 1.0d0 c c solve l*v = y c do 140 k = 1, n l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t lm = min0(ml,n-k) if (k .lt. n) call daxpy(lm,t,abd(m+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 130 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 130 continue 140 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) ynorm = s*ynorm c c solve u*z = w c do 160 kb = 1, n k = n + 1 - kb if (dabs(z(k)) .le. dabs(abd(m,k))) go to 150 s = dabs(abd(m,k))/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 150 continue if (abd(m,k) .ne. 0.0d0) z(k) = z(k)/abd(m,k) if (abd(m,k) .eq. 0.0d0) z(k) = 1.0d0 lm = min0(k,m) - 1 la = m - lm lz = k - lm t = -z(k) call daxpy(lm,t,abd(la,k),1,z(lz),1) 160 continue c make znorm = 1.0 s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) ynorm = s*ynorm c if (anorm .ne. 0.0d0) rcond = ynorm/anorm if (anorm .eq. 0.0d0) rcond = 0.0d0 return end subroutine dgeco(a,lda,n,ipvt,rcond,z) integer lda,n,ipvt(1) double precision a(lda,1),z(1) double precision rcond c c dgeco factors a double precision matrix by gaussian elimination c and estimates the condition of the matrix. c c if rcond is not needed, dgefa is slightly faster. c to solve a*x = b , follow dgeco by dgesl. c to compute inverse(a)*c , follow dgeco by dgesl. c to compute determinant(a) , follow dgeco by dgedi. c to compute inverse(a) , follow dgeco by dgedi. c c on entry c c a double precision(lda, n) c the matrix to be factored. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c on return c c a an upper triangular matrix and the multipliers c which were used to obtain it. c the factorization can be written a = l*u where c l is a product of permutation and unit lower c triangular matrices and u is upper triangular. c c ipvt integer(n) c an integer vector of pivot indices. c c rcond double precision c an estimate of the reciprocal condition of a . c for the system a*x = b , relative perturbations c in a and b of size epsilon may cause c relative perturbations in x of size epsilon/rcond . c if rcond is so small that the logical expression c 1.0 + rcond .eq. 1.0 c is true, then a may be singular to working c precision. in particular, rcond is zero if c exact singularity is detected or the estimate c underflows. c c z double precision(n) c a work vector whose contents are usually unimportant. c if a is close to a singular matrix, then z is c an approximate null vector in the sense that c norm(a*z) = rcond*norm(a)*norm(z) . c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c linpack dgefa c blas daxpy,ddot,dscal,dasum c fortran dabs,dmax1,dsign c c internal variables c double precision ddot,ek,t,wk,wkm double precision anorm,s,dasum,sm,ynorm integer info,j,k,kb,kp1,l c c c compute 1-norm of a c anorm = 0.0d0 do 10 j = 1, n anorm = dmax1(anorm,dasum(n,a(1,j),1)) 10 continue c c factor c call dgefa(a,lda,n,ipvt,info) c c rcond = 1/(norm(a)*(estimate of norm(inverse(a)))) . c estimate = norm(z)/norm(y) where a*z = y and trans(a)*y = e . c trans(a) is the transpose of a . the components of e are c chosen to cause maximum local growth in the elements of w where c trans(u)*w = e . the vectors are frequently rescaled to avoid c overflow. c c solve trans(u)*w = e c ek = 1.0d0 do 20 j = 1, n z(j) = 0.0d0 20 continue do 100 k = 1, n if (z(k) .ne. 0.0d0) ek = dsign(ek,-z(k)) if (dabs(ek-z(k)) .le. dabs(a(k,k))) go to 30 s = dabs(a(k,k))/dabs(ek-z(k)) call dscal(n,s,z,1) ek = s*ek 30 continue wk = ek - z(k) wkm = -ek - z(k) s = dabs(wk) sm = dabs(wkm) if (a(k,k) .eq. 0.0d0) go to 40 wk = wk/a(k,k) wkm = wkm/a(k,k) go to 50 40 continue wk = 1.0d0 wkm = 1.0d0 50 continue kp1 = k + 1 if (kp1 .gt. n) go to 90 do 60 j = kp1, n sm = sm + dabs(z(j)+wkm*a(k,j)) z(j) = z(j) + wk*a(k,j) s = s + dabs(z(j)) 60 continue if (s .ge. sm) go to 80 t = wkm - wk wk = wkm do 70 j = kp1, n z(j) = z(j) + t*a(k,j) 70 continue 80 continue 90 continue z(k) = wk 100 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c c solve trans(l)*y = w c do 120 kb = 1, n k = n + 1 - kb if (k .lt. n) z(k) = z(k) + ddot(n-k,a(k+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 110 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) 110 continue l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t 120 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) c ynorm = 1.0d0 c c solve l*v = y c do 140 k = 1, n l = ipvt(k) t = z(l) z(l) = z(k) z(k) = t if (k .lt. n) call daxpy(n-k,t,a(k+1,k),1,z(k+1),1) if (dabs(z(k)) .le. 1.0d0) go to 130 s = 1.0d0/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 130 continue 140 continue s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) ynorm = s*ynorm c c solve u*z = v c do 160 kb = 1, n k = n + 1 - kb if (dabs(z(k)) .le. dabs(a(k,k))) go to 150 s = dabs(a(k,k))/dabs(z(k)) call dscal(n,s,z,1) ynorm = s*ynorm 150 continue if (a(k,k) .ne. 0.0d0) z(k) = z(k)/a(k,k) if (a(k,k) .eq. 0.0d0) z(k) = 1.0d0 t = -z(k) call daxpy(k-1,t,a(1,k),1,z(1),1) 160 continue c make znorm = 1.0 s = 1.0d0/dasum(n,z,1) call dscal(n,s,z,1) ynorm = s*ynorm c if (anorm .ne. 0.0d0) rcond = ynorm/anorm if (anorm .eq. 0.0d0) rcond = 0.0d0 return end subroutine dgedi(a,lda,n,ipvt,det,work,job) integer lda,n,ipvt(1),job double precision a(lda,1),det(2),work(1) c c dgedi computes the determinant and inverse of a matrix c using the factors computed by dgeco or dgefa. c c on entry c c a double precision(lda, n) c the output from dgeco or dgefa. c c lda integer c the leading dimension of the array a . c c n integer c the order of the matrix a . c c ipvt integer(n) c the pivot vector from dgeco or dgefa. c c work double precision(n) c work vector. contents destroyed. c c job integer c = 11 both determinant and inverse. c = 01 inverse only. c = 10 determinant only. c c on return c c a inverse of original matrix if requested. c otherwise unchanged. c c det double precision(2) c determinant of original matrix if requested. c otherwise not referenced. c determinant = det(1) * 10.0**det(2) c with 1.0 .le. dabs(det(1)) .lt. 10.0 c or det(1) .eq. 0.0 . c c error condition c c a division by zero will occur if the input factor contains c a zero on the diagonal and the inverse is requested. c it will not occur if the subroutines are called correctly c and if dgeco has set rcond .gt. 0.0 or dgefa has set c info .eq. 0 . c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c blas daxpy,dscal,dswap c fortran dabs,mod c c internal variables c double precision t double precision ten integer i,j,k,kb,kp1,l,nm1 c c c compute determinant c if (job/10 .eq. 0) go to 70 det(1) = 1.0d0 det(2) = 0.0d0 ten = 10.0d0 do 50 i = 1, n if (ipvt(i) .ne. i) det(1) = -det(1) det(1) = a(i,i)*det(1) c ...exit if (det(1) .eq. 0.0d0) go to 60 10 if (dabs(det(1)) .ge. 1.0d0) go to 20 det(1) = ten*det(1) det(2) = det(2) - 1.0d0 go to 10 20 continue 30 if (dabs(det(1)) .lt. ten) go to 40 det(1) = det(1)/ten det(2) = det(2) + 1.0d0 go to 30 40 continue 50 continue 60 continue 70 continue c c compute inverse(u) c if (mod(job,10) .eq. 0) go to 150 do 100 k = 1, n a(k,k) = 1.0d0/a(k,k) t = -a(k,k) call dscal(k-1,t,a(1,k),1) kp1 = k + 1 if (n .lt. kp1) go to 90 do 80 j = kp1, n t = a(k,j) a(k,j) = 0.0d0 call daxpy(k,t,a(1,k),1,a(1,j),1) 80 continue 90 continue 100 continue c c form inverse(u)*inverse(l) c nm1 = n - 1 if (nm1 .lt. 1) go to 140 do 130 kb = 1, nm1 k = n - kb kp1 = k + 1 do 110 i = kp1, n work(i) = a(i,k) a(i,k) = 0.0d0 110 continue do 120 j = kp1, n t = work(j) call daxpy(n,t,a(1,j),1,a(1,k),1) 120 continue l = ipvt(k) if (l .ne. k) call dswap(n,a(1,k),1,a(1,l),1) 130 continue 140 continue 150 continue return end