(PC-12xx~14xx) Laguerre method
The classic "sure-shot" Laguerre root finding method. C and BASIC implementations are included with comments. This version of the Laguerre method is implemented with the same optimizations applied the Householder root finder posted previously. A quick test shows that Laguerre solves roots 15% faster than the Householder method on a Sharp PC-1475 for the five examples listed below, while requiring the same number of iterations. Results may differ for other machines.
LAGUERRE POLYNOMIAL ROOT FINDER
3th order convergence, worst-case linear convergence for roots with multiplicities
Optimized implementations in C and BASIC:
- Finds complex roots z of polynomial p(z)
- Customizable floating point accuracy MACHEPS
- Uses a novel root starting point to improve performance
- Prevents overflow/nan/inf
- Prevents loss of precision when possible, e.g. in reciprocals
- Tested with a large sample of polynomials
See also: https://en.wikipedia.org/wiki/Laguerre's_method
W.H. Press et al. Numerical Recipes FORTRAN 2ed
David Goldberg. What Every Computer Scientist Should Know About Floating Point, ACM Computing Surveys, 1991
J.M. McNamee. Numerical Methods for Roots of Polynomials, Part I & II, Elsevier, Kidlington, Oxford 2009
For SHARP PC-12xx to 14xx
To use with SHARP PC models with a single display line:
- remove CURSOR and CLS
- replace PRINT with PAUSE except at line 60
1,761 bytes BASIC image (PC-1350)
Degree: max polynomial degree limited by time and memory only
Example:
(Solve x^3-2x^2-x+2=0)
[DEF-A]
N=3 A3=1 A2=-2 A1=-1 A0=2
CALCULATING...
RE IM
2. 0.
-1. 0.
1. 0.
(Solve x^3+3x^2+x+3=0)
[DEF-A]
N=3 A3=1 A2=3 A1=1 A0=3
CALCULATING...
RE IM
0. 1.
0. -1.
-3. 0.
(Solve x^4+2x^3-13x^2-14x+24=0)
[DEF-A]
N=4 A4=1 A3=2 A2=-13 A1=-14 A0=24
CALCULATING...
RE IM
3. 0.
-4. 0.
-2. 0.
1. 0.
(Solve x^4+2999x^3-10003e3x^2-2399e7x+24e9=0)
[DEF-A]
N=4 A4=1 A3=2999 A2=-10003E3 A1=-2399E7 A0=24E9
CALCULATING...
RE IM
3000. 0.
-4000. 0.
-2000. 0.
1. 0.
(Solve 5x^6-45x^5+225x^4-425x^3+170x^2+370x-500=0)
[DEF-A]
N=6 A6=5 A5=-45 A4=225 A3=-425 A2=170 A1=370 A0=-500
CALCULATING...
RE IM
3. 4.
3. -4.
2. 0.
-1. 0.
1. -1.
1. 1.
Code:
' LAGUERRE POLYNOMIAL ROOT FINDER
' 3th order convergence, worst-case linear convergence for roots with multiplicities
'
' For SHARP PC-12xx to 14xx by Robert van Engelen
' To use with SHARP PC models with a single display line:
' - remove CURSOR and CLS
' - replace PRINT with PAUSE except at line 60
'
' Optimized implementations in C and BASIC:
' - Finds complex roots z of polynomial p(z)
' - Customizable floating point accuracy MACHEPS
' - Uses a novel root starting point to improve performance
' - Prevents overflow/nan/inf
' - Prevents loss of precision when possible, e.g. in reciprocals
' - Tested with a large sample of polynomials
'
' See also: https://en.wikipedia.org/wiki/Laguerre's_method
' W.H. Press et al. Numerical Recipes FORTRAN 2ed
' David Goldberg. What Every Computer Scientist Should Know About Floating Point, ACM Computing Surveys, 1991
' J.M. McNamee. Numerical Methods for Roots of Polynomials, Part I & II, Elsevier, Kidlington, Oxford 2009
' Algorithm to find complex roots of a polynomial with complex coefficients:
' // define MachEps machine epsilon precision, pow(FLT_RADIX, -DBL_MANT_DIG + 1) for double precision
' #define MACHEPS 1E-16
' // squared magnitude of complex z
' #define cnorm(z) (creal(z) * creal(z) + cimag(z) * cimag(z))
' void Laguerre(int n, const complex double a[], complex double z[]) {
' int i, m;
' double e, t;
' complex double *b, p, p1, p2, r, h, g, u, v;
' // find the trailing zero coefficients and remove them
' for (m = n; m > 1 && a[m] == 0; --m)
' z[m-1] = 0;
' // leading coefficient cannot be zero
' if (m < 1 || a[0] == 0)
' return -1;
' // copy coefficients to deflate the polynomial
' b = (complex double*)malloc((m + 1) * sizeof(complex double));
' memcpy(b, a, (m + 1) * sizeof(complex double));
' // find the complex roots z[m-1] to z[2]
' while (m > 2) {
' // pick a suitable stating point for the root search
' r = sqrt(0.5);
' if (b[m-1] != 0)
' r *= -b[m] / b[m-1];
' // my novel idea (invention) to improve the complex root starting point for faster convergence
' if (cimag(r) == 0)
' r = CMPLX(creal(r), creal(r)/m);
' // root modulus error tolerance
' e = 6 * m * cabs(b[m]) * MACHEPS;
' // loop to compute root z[m-1]
' do {
' // compute p(z), p'(z), p"(z) using Horner's rule
' p = b[0];
' p1 = 0;
' p2 = 0;
' for (i = 1; i <= m; ++i) {
' p2 = p2 * r + p1;
' p1 = p1 * r + p;
' p = p * r + b[i];
' }
' // if |p(z)| is (close to) zero then stop searching; using triangle inequality to prevent overflow
' if (fabs(creal(p)) < e && fabs(cimag(p)) < e && cnorm(p) < e*e)
' break;
' // compute G = p'(z)/p(z), H = G^2-p"(z)/p(z)
' g = p1/p;
' u = g*g;
' h = u - p2/p;
' // compute h = m/(G+/-sqrt((m-1)*(m*H-G^2))) with maximal denominator magnitude
' v = csqrt((m-1) * (m*h - u));
' u = g+v;
' v = g-v;
' if (cnorm(u) < cnorm(v))
' u = v;
' h = m/u;
' r -= h;
' }
' while (cnorm(h) > (cnorm(r) + 1) * MACHEPS*MACHEPS);
' if (fabs(cimag(r)) < MACHEPS) {
' // |z.im| < MachEps, consider root is real
' r = creal(r);
' }
' else {
' // compute p(z.re) using Horner's rule
' t = cnorm(p);
' p = b[0];
' for (i = 1; i <= m; ++i)
' p = p * creal(r) + b[i];
' // if |p(z.re)| <= |p(z)| then root is real
' if (cnorm(p) <= t)
' r = creal(r);
' }
' // save the complex root
' z[m-1] = r;
' // forward deflate the polynomial
' v = b[0];
' for (i = 1; i < m; ++i)
' v = b[i] += v * r;
' --m;
' }
' // compute the roots for the remaining m=1 and m=2 cases
' if (m == 1) {
' // linear case
' z[0] = -b[1] / b[0];
' }
' else if (b[1] == 0) {
' // quadratic case when b=0
' z[0] = csqrt(-b[2] / b[0]);
' z[1] = -z[0];
' }
' else {
' // quadratic formula without potential catastrophic cancellation
' v = b[1] * b[1] - 4 * b[0] * b[2];
' if (creal(v) <= 0 || creal(b[1]) < 0)
' z[0] = (-b[1] + csqrt(v)) / (2 * b[0]);
' else
' z[0] = (2 * b[2]) / (-b[1] - csqrt(v));
' z[1] = b[2] / (b[0] * z[0]);
' }
' free(b);
' }
'
' Algorithm to find complex roots of a polynomial with real coefficients, using separate real and imaginary variables:
' // define MachEps machine epsilon precision, pow(FLT_RADIX, -DBL_MANT_DIG + 1) for double precision
' #define MACHEPS 1E-16
' // squared magnitude of complex (zr,zi)
' #define norm(zr, zi) ((zr) * (zr) + (zi) * (zi))
' void laguerre(int n, const double a[], double zr[], double zi[]) {
' int i, m;
' double *b, e, k1, k2, pr, pi, pr1, pi1, pr2, pi2, rr, ri, hr, hi, gr, gi, ur, ui, vr, vi;
' // find the trailing zero coefficients and remove them
' for (m = n; m > 1 && a[m] == 0; --m)
' zr[m-1] = zi[n-1] = 0;
' // leading coefficient cannot be zero
' if (m < 1 || a[0] == 0)
' return -1;
' // copy coefficients to deflate the polynomial
' b = (double*)malloc((m + 1) * sizeof(double));
' memcpy(b, a, (m + 1) * sizeof(double));
' // loop to find the complex roots z[m-1] to z[2]
' while (m > 2) {
' // pick a suitable stating point for the root search
' rr = sqrt(0.5);
' if (b[m-1] != 0)
' rr *= -b[m] / b[m-1];
' // my novel idea (invention) to improve the complex root starting point for faster convergence
' ri = rr/m;
' // root modulus error tolerance
' e = 6 * m * fabs(b[m]) * MACHEPS;
' // loop to compute root z[m-1]
' do {
' // compute p(z), p'(z), p"(z) using Horner's rule
' pr = b[0];
' pi = 0;
' pr1 = 0;
' pi1 = 0;
' pr2 = 0;
' pi2 = 0;
' for (i = 1; i <= m; ++i) {
' // compute p2 = p2 * r + p1
' k1 = pr2;
' pr2 = pr2 * rr - pi2 * ri + pr1;
' pi2 = pi2 * rr + k1 * ri + pi1;
' // compute p1 = p1 * r + p
' k1 = pr1;
' pr1 = pr1 * rr - pi1 * ri + pr;
' pi1 = pi1 * rr + k1 * ri + pi;
' // compute p = p * r + b[i]
' k1 = pr;
' pr = pr * rr - pi * ri + b[i];
' pi = pi * rr + k1 * ri;
' }
' // if |p(z)| is (close to) zero then stop searching; using triangle inequality to prevent overflow
' if (fabs(pr) < e && fabs(pi) < e && norm(pr, pi) < e*e)
' break;
' // compute G = p'(z)/p(z) and p"(z)/p(z)
' if (fabs(pr) >= fabs(pi)) {
' k1 = pi/pr;
' k2 = k1*pi + pr;
' gr = (pr1 + k1*pi1) / k2;
' gi = (pi1 - k1*pr1) / k2;
' hr = (pr2 + k1*pi2) / k2;
' hi = (pi2 - k1*pr2) / k2;
' }
' else {
' k1 = pr/pi;
' k2 = k1*pr + pi;
' gr = (k1*pr1 + pi1) / k2;
' gi = (k1*pi1 - pr1) / k2;
' hr = (k1*pr2 + pi2) / k2;
' hi = (k1*pi2 - pr2) / k2;
' }
' // compute G^2
' ur = gr*gr - gi*gi;
' ui = 2*gr*gi;
' // compute H = G^2-p"(z)/p(z)
' hr = ur-hr;
' hi = ui-hi;
' // compute v = sqrt((m-1)*(m*H-G^2)) without potential intermediate overflow
' hr = (m-1) * (m*hr - ur);
' hi = (m-1) * (m*hi - ui);
' if (hi == 0) {
' if (hr < 0) {
' hi = sqrt(-hr);
' hr = 0;
' }
' else {
' hr = sqrt(hr);
' }
' }
' else {
' if (fabs(hr) < fabs(hi)) {
' k1 = hr/hi;
' k1 = fabs(hi) * sqrt(1 + k1*k1);
' }
' else {
' k1 = hi/hr;
' k1 = fabs(hr) * sqrt(1 + k1*k1);
' }
' vr = sqrt((k1+hr) / 2);
' vi = copysign(sqrt((k1-hr) / 2), hi);
' }
' // compute h = m/(G+/-sqrt((m-1)*(m*H-G^2))) with maximal denominator magnitude
' ur = gr+vr;
' ui = gi+vi;
' vr = gr-vr;
' vi = gi-vi;
' if (norm(ur, ui) < norm(vr, vi)) {
' ur = vr;
' ui = vi;
' }
' if (fabs(ur) >= fabs(ui)) {
' k1 = ui/ur;
' k2 = k1*ui + ur;
' hr = m / k2;
' hi = -k1*m / k2;
' }
' else {
' k1 = ur/ui;
' k2 = k1*ur + ui;
' hr = k1*m / k2;
' hi = -m / k2;
' }
' rr -= hr;
' ri -= hi;
' }
' while (norm(hr, hi) > (norm(rr, ri) + 1) * MACHEPS*MACHEPS);
' // if |z.im| < MachEps then root is real
' if (fabs(ri) < MACHEPS) {
' ri = 0;
' }
' else {
' // compute p(z.re) using Horner's rule
' k1 = b[0];
' for (i = 1; i <= m; ++i)
' k1 = k1 * rr + b[i];
' // if |p(z.re)| <= |p(z)| then root is real
' if (k1*k1 <= norm(pr, pi))
' ri = 0;
' }
' // save the complex root
' zr[m-1] = rr;
' zi[m-1] = ri;
' if (ri == 0) {
' // root is real; deflate the polynomial
' k1 = b[0];
' for (i = 1; i < m; ++i)
' k1 = b[i] += k1 * rr;
' }
' else {
' // root is complex; add conjugate root and deflate by quadratic
' --m;
' zr[m-1] = rr;
' zi[m-1] = -ri;
' k1 = -2 * rr;
' k2 = rr*rr + ri*ri;
' b[1] -= k1 * b[0];
' for (i = 2; i < m; ++i)
' b[i] -= k1 * b[i-1] + k2 * b[i-2];
' }
' --m;
' }
' // compute the roots for the remaining m=1 and m=2 cases
' if (m == 1) {
' // linear case
' zr[0] = -b[1] / b[0];
' zi[0] = 0;
' }
' else if (b[1] == 0) {
' // quadratic case when b=0
' k1 = -b[2] / b[0];
' if (k1 > 0) {
' zr[0] = sqrt(k1);
' zi[0] = 0;
' zr[1] = -zr[0];
' zi[1] = 0;
' }
' else {
' zr[0] = 0;
' zi[0] = sqrt(-k1);
' zr[1] = 0;
' zi[1] = -zi[0];
' }
' }
' else {
' // quadratic formula without potential catastrophic cancellation
' k1 = b[1] * b[1] - 4 * b[0] * b[2];
' if (k1 >= 0) {
' if (b[1] > 0) {
' k2 = -b[1] - sqrt(k1);
' zr[0] = 2 * b[2] / k2;
' zi[0] = 0;
' zr[1] = 0.5 * k2 / b[0];
' zi[1] = 0;
' }
' else {
' k2 = -b[1] + sqrt(k1);
' zr[0] = 0.5 * k2 / b[0];
' zi[0] = 0;
' zr[1] = 2 * b[2] / k2;
' zi[1] = 0;
' }
' }
' else {
' zr[0] = -0.5 * b[1] / b[0];
' zi[0] = 0.5 * sqrt(-k1) / fabs(b[0]);
' zr[1] = zr[0];
' zi[1] = -zi[0];
' }
' }
' free(b);
' }
' VARIABLES
' N degree
' A(27..27+N) N+1 coefficients indexed from highest to lowest
' A(28+N..28+2N) N+1 copied coefficients used in deflation
' A(O..O+N-1) N roots real parts
' A(Z..Z+N-1) N roots imag parts
' O array offset in A() with the real parts of the roots
' Z array offset in A() with the imag parts of the roots
' I loop iterator
' L array offset in A() with copied coefficients used in deflation
' M internal adjusted degree M<=N
' A,B real and imag parts of root z
' P,Q real and imag parts of p(z)
' C,D real and imag parts of p'(z)
' R,S real and imag parts of p"(z)
' E eps = 6*m*|a[m]|*MachEps
' F,G real and imag parts of g
' H,K real and imag parts of h
' U,X real and imag parts of u
' V,Y real and imag parts of v
' T,W scratch
' driver program
10 "A" CLS: WAIT 0: INPUT "N=";N: IF N<2 GOTO 10
20 FOR I=0 TO N: CLS: PRINT "A";STR$(N-I);"=";A(27+I): CURSOR 24: INPUT A(27+I)
30 NEXT I
40 CLS: PRINT "CALCULATING...": GOSUB "LAG"
50 PRINT "RE","IM": WAIT
60 FOR I=0 TO N-1: PRINT A(O+I),A(Z+I): NEXT I
70 END
' polynomial root finder: N degree, A(27..27+N) coefficients
100 "LAG" L=28+N,O=L+N+1,Z=O+N,M=N
' optimized: remove leading zero coefficients (trailing A)
110 IF M>1 IF A(27+M)=0 LET M=M-1,A(O+M)=0,A(Z+M)=0: GOTO 110
120 IF M<1 OR A(27)=0 RETURN
' copy coefficients to deflate the polynomial
130 FOR I=0 TO M: A(L+I)=A(27+I): NEXT I
' loop to find the complex roots z[m-1] to z[2]
140 IF M<3 GOTO 440
150 FOR M=M TO 3 STEP -1: CURSOR 24: PRINT "ROOT ";M
160 A=SQR .5: IF A(L+M-1)<>0 LET A=-A*A(L+M)/A(L+M-1)
170 B=A/M,E=6E-10*M*ABS A(L+M)
' loop to compute root z[m-1]
' compute p(z), p'(z), p"(z) using Horner's rule
180 P=A(L),Q=0,C=0,D=0,R=0,S=0
190 FOR I=1 TO M
200 T=R,R=R*A-S*B+C,S=S*A+T*B+D,T=C,C=C*A-D*B+P,D=D*A+T*B+Q
210 T=P,P=P*A-Q*B+A(L+I),Q=Q*A+T*B
220 NEXT I
' if |p(z)| is (close to) zero then stop searching; using triangle inequality to prevent overflow
230 IF ABS P<E IF ABS Q<E IF P*P+Q*Q < E*E GOTO 360
' compute G = p'(z)/p(z) and p"(z)/p(z)
240 IF ABS P<ABS Q GOTO 255
250 T=Q/P,W=T*Q+P,F=(C+T*D)/W,G=(D-T*C)/W,H=(R+T*S)/W,K=(S-T*R)/W: GOTO 260
' note: assignments moved from line 240 (after the IF) to line 255 with GOTO 260 removed, to make it fit on some Sharp PC
255 T=P/Q,W=T*P+Q,F=(T*C+D)/W,G=(T*D-C)/W,H=(T*R+S)/W,K=(T*S-R)/W
' compute G^2 and H = G^2-p"(z)/p(z) and v = sqrt((m-1)*(m*H-G^2)) without potential intermediate overflow
260 U=F*F-G*G,X=2*F*G,H=U-H,K=X-K,H=(M-1)*(M*H-U),K=(M-1)*(M*K-X)
270 IF K=0 LET T=SQR ABS H,H=(H>0)*T,K=(H<0)*T: GOTO 310
280 IF ABS H<ABS K LET T=H/K,T=ABS K*SQR(1+T*T): GOTO 300
290 T=K/H,T=ABS H*SQR(1+T*T)
300 V=SQR(.5*(T+H)),Y=SGN K*SQR(.5*(T-H))
' compute h = m/(G+/-sqrt((m-1)*(m*H-G^2))) with maximal denominator magnitude
310 U=F+V,X=G+Y,V=F-V,Y=G-Y: IF U*U+X*X<V*V+Y*Y LET U=V,X=Y
320 IF ABS U<ABS X LET T=U/X,W=T*U+X,H=T*M/W,K=-M/W: GOTO 340
330 T=X/U,W=T*X+U,H=M/W,K=-T*M/W
340 A=A-H,B=B-K
' loop while |h|^2 > (|r|^2 + 1) * MachEps
350 IF H*H+K*K>(A*A+B*B+1)*1E-20 GOTO 180
' if |z.im| < MachEps then root is real
360 IF ABS B<1E-10 LET B=0: GOTO 390
' compute p(z.re) using Horner's rule
370 T=A(L): FOR I=1 TO M: T=T*A+A(L+I): NEXT I
' if |p(z.re)| <= |p(z)| then root is real
380 IF T*T<=P*P+Q*Q LET B=0
390 A(O+M-1)=A,A(Z+M-1)=B
' if root is real then deflate the polynomial
400 IF B=0 LET T=A(L): FOR I=1 TO M-1: T=A(L+I)+T*A,A(L+I)=T: NEXT I: GOTO 430
' root is complex; add conjugate root and deflate by quadratic
410 M=M-1:A(O+M-1)=A,A(Z+M-1)=-B,T=-2*A,W=A*A+B*B,A(L+1)=A(L+1)-T*A(L)
420 FOR I=2 TO M-1: A(L+I)=A(L+I)-T*A(L+I-1)-W*A(L+I-2): NEXT I
430 NEXT M
' compute the roots for the remaining m=1 and m=2 cases
440 IF M=1 LET A(O)=-A(L+1)/A(L),A(Z)=0: RETURN
450 IF A(L+1)<>0 GOTO 480
460 T=-A(L+2)/A(L): IF T>0 LET A(O)=SQR T,A(Z)=0,A(O+1)=-A(O),A(Z+1)=0: RETURN
470 A(O)=0,A(Z)=SQR -T,A(O+1)=0,A(Z+1)=-A(Z): RETURN
' quadratic formula without potential catastrophic cancellation
480 T=A(L+1),W=T*T-4*A(L)*A(L+2)
490 IF W<0 LET A(O)=-.5*T/A(L),A(Z)=.5*SQR(-W)/ABS A(L),A(O+1)=A(O),A(Z+1)=-A(Z): RETURN
500 IF T>0 LET W=-T-SQR W,A(O)=2*A(L+2)/W,A(Z)=0,A(O+1)=.5*W/A(L),A(Z+1)=0: RETURN
510 W=-T+SQR W,A(O)=.5*W/A(L),A(Z)=0,A(O+1)=2*A(L+2)/W,A(Z+1)=0: RETURN
"I count on old friends to remain rational"
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