2 x^2+y^2-4; x*y-1; ROOT COUNTS : total degree : 4 1-homogeneous Bezout number : 4 with partition : { x y } general linear-product Bezout number : 4 based on the set structure : { x y }{ x y } { x }{ y } mixed volume : 4 stable mixed volume : 4 TIMING INFORMATION for Root Counting The elapsed time in seconds was 0.002645000 = 0h 0m 0s 3ms User time in seconds was 0.000525000 = 0h 0m 0s 1ms System CPU time in seconds was 0.002120000 = 0h 0m 0s 2ms Non-I/O page faults was 3 I/O page faults was 78 Signals delivered was 0 Swaps was 0 Total context switches was 18 START SYSTEM BASED ON TOTAL DEGREE : 2 +1*x^2 +(-7.17217976217691E-01 - 6.96848889351343E-01*i); +1*y^2 +( 5.69940499123200E-01 - 8.21685966448982E-01*i); START SOLUTIONS : 4 2 =========================================================================== solution 1 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : -9.26611562688943E-01 -3.76019962091316E-01 y : -4.63713004387843E-01 -8.85985468030712E-01 == err : 0.000E+00 = rco : 1.000E+00 = res : 0.000E+00 == solution 2 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : -9.26611562688943E-01 -3.76019962091316E-01 y : 4.63713004387844E-01 8.85985468030712E-01 == err : 0.000E+00 = rco : 1.000E+00 = res : 0.000E+00 == solution 3 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : 9.26611562688943E-01 3.76019962091316E-01 y : -4.63713004387843E-01 -8.85985468030712E-01 == err : 0.000E+00 = rco : 1.000E+00 = res : 0.000E+00 == solution 4 : t : 0.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : 9.26611562688943E-01 3.76019962091316E-01 y : 4.63713004387844E-01 8.85985468030712E-01 == err : 0.000E+00 = rco : 1.000E+00 = res : 0.000E+00 == TIMING INFORMATION for Construction of Start System The elapsed time in seconds was 0.000323000 = 0h 0m 0s 0ms User time in seconds was 0.000063000 = 0h 0m 0s 0ms System CPU time in seconds was 0.000260000 = 0h 0m 0s 0ms Non-I/O page faults was 0 I/O page faults was 6 Signals delivered was 0 Swaps was 0 Total context switches was 2 HOMOTOPY PARAMETERS : k : 2 gamma : 8.41612399095819E-01 -5.40082002744194E-01 t : 1.00000000000000E+00 0.00000000000000E+00 no projective transformation ****************** CURRENT CONTINUATION PARAMETERS ***************** GLOBAL MONITOR : 1. the condition of the homotopy : 0 2. number of paths tracked simultaneously : 1 3. maximum number of steps along a path : 500 4. distance from target to start end game : 1.000E-01 5. order of extrapolator in end game : 0 6. maximum number of re-runs : 1 STEP CONTROL (PREDICTOR) : along path : end game 7: 8. type ( x:Cub,t:Rea ):( x:Cub,t:Rea ) : 8 : 8 9:10. minimum step size : 1.000E-06 : 1.000E-08 11:12. maximum step size : 1.000E-01 : 1.000E-02 13:14. reduction factor for step size : 7.000E-01 : 5.000E-01 15:16. expansion factor for step size : 1.250E+00 : 1.100E+00 17:18. expansion threshold : 1 : 3 PATH CLOSENESS (CORRECTOR) : along path : end game 19:20. maximum number of iterations : 3 : 3 21:22. relative precision for residuals : 1.000E-09 : 1.000E-11 23:24. absolute precision for residuals : 1.000E-09 : 1.000E-11 25:26. relative precision for corrections : 1.000E-09 : 1.000E-11 27:28. absolute precision for corrections : 1.000E-09 : 1.000E-11 SOLUTION TOLERANCES : along path : end game 29:30. inverse condition of Jacobian : 1.000E-04 : 1.000E-12 31:32. clustering of solutions : 1.000E-04 : 1.000E-12 33:34. solution at infinity : 1.000E+08 : 1.000E+12 ******************************************************************** THE SOLUTIONS : 4 2 =========================================================================== == 1 = #step : 10 #fail : 0 #iter : 31 = regular solution == t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 Length of path : 4.55615676505241E-01 the solution for t : x : -1.93185165257814E+00 4.48415508583941E-44 y : -5.17638090205041E-01 0.00000000000000E+00 == err : 9.065E-17 = rco : 5.774E-01 = res : 2.220E-16 == == 2 = #step : 24 #fail : 4 #iter : 67 = regular solution == t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 Length of path : 4.00645258285465E+00 the solution for t : x : 5.17638090205041E-01 3.40156822331300E-27 y : 1.93185165257814E+00 -3.30690491468648E-27 == err : 1.118E-13 = rco : 5.774E-01 = res : 1.388E-16 == == 3 = #step : 24 #fail : 4 #iter : 67 = regular solution == t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 Length of path : 4.00645258285465E+00 the solution for t : x : -5.17638090205041E-01 -3.40156822331300E-27 y : -1.93185165257814E+00 3.30690491468648E-27 == err : 1.118E-13 = rco : 5.774E-01 = res : 1.388E-16 == == 4 = #step : 10 #fail : 0 #iter : 31 = regular solution == t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 Length of path : 4.55615676505240E-01 the solution for t : x : 1.93185165257814E+00 -4.48415508583941E-44 y : 5.17638090205041E-01 0.00000000000000E+00 == err : 9.065E-17 = rco : 5.774E-01 = res : 2.220E-16 == == #regu : 4 = #sing : 0 = #clus : 0 = #infi : 0 = #fail : 0 == TIMING INFORMATION for continuation The elapsed time in seconds was 0.001862000 = 0h 0m 0s 2ms User time in seconds was 0.000768000 = 0h 0m 0s 1ms System CPU time in seconds was 0.001094000 = 0h 0m 0s 1ms Non-I/O page faults was 1 I/O page faults was 26 Signals delivered was 0 Swaps was 0 Total context switches was 9 THE SOLUTIONS : 4 2 =========================================================================== solution 1 : start residual : 3.331E-16 #iterations : 1 success t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : -1.93185165257814E+00 -4.36015087616835E-106 y : -5.17638090205042E-01 4.36015087616835E-106 == err : 2.355E-16 = rco : 5.774E-01 = res : 2.220E-16 = real regular == solution 2 : start residual : 2.498E-16 #iterations : 1 success t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : 5.17638090205041E-01 -1.25672792711622E-88 y : 1.93185165257814E+00 0.00000000000000E+00 == err : 1.700E-16 = rco : 5.774E-01 = res : 2.498E-16 = real regular == solution 3 : start residual : 2.498E-16 #iterations : 1 success t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : -5.17638090205041E-01 1.25672792711622E-88 y : -1.93185165257814E+00 0.00000000000000E+00 == err : 1.700E-16 = rco : 5.774E-01 = res : 2.498E-16 = real regular == solution 4 : start residual : 3.331E-16 #iterations : 1 success t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x : 1.93185165257814E+00 4.36015087616835E-106 y : 5.17638090205042E-01 -4.36015087616835E-106 == err : 2.355E-16 = rco : 5.774E-01 = res : 2.220E-16 = real regular == =========================================================================== A list of 4 solutions has been refined : Number of regular solutions : 4. Number of singular solutions : 0. Number of real solutions : 4. Number of complex solutions : 0. Number of clustered solutions : 0. Number of solutions at infinity : 0. Number of failures : 0. =========================================================================== Frequency tables for correction, residual, and condition numbers : FreqCorr : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 : 4 FreqResi : 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 : 4 FreqCond : 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 4 Small correction terms and residuals counted to the right. Well conditioned and distinct roots counted to the left. TIMING INFORMATION for Solving the polynomial system The elapsed time in seconds was 0.008745000 = 0h 0m 0s 9ms User time in seconds was 0.002738000 = 0h 0m 0s 3ms System CPU time in seconds was 0.006007000 = 0h 0m 0s 6ms Non-I/O page faults was 13 I/O page faults was 198 Signals delivered was 0 Swaps was 0 Total context switches was 52 --------------------------------------------------------------------- | TIMING INFORMATION SUMMARY | --------------------------------------------------------------------- | root counts | start system | continuation | total time | --------------------------------------------------------------------- | 0h 0m 0s 1ms | 0h 0m 0s 0ms | 0h 0m 0s 1ms | 0h 0m 0s 3ms | --------------------------------------------------------------------- PHC ran from 1 January 2019, 15:42:47 till 1 January 2019, 15:42:47. The total elapsed wall clock time is 0.039 seconds = 39 milliseconds. No multitasking was used in this run. Seed used in random number generators : 7741. PHCv2.4.63 released 2018-12-28