Macaulay 2, version 0.9.95 with packages: Classic, Core, Elimination, IntegralClosure, LLLBases, Parsing, PrimaryDecomposition, SchurRings, TangentCone i1 : S:=QQ[x,y] o1 = QQ [x, y] o1 : PolynomialRing i2 : loadPackage "SOS" ... ii3 : f=x^2*y^4+x^4*y^2-x^2*y^2+1 4 2 2 4 2 2 oo3 = x y + x y - x y + 1 oo3 : QQ [x, y] ii8 : getSOS(f^3) ----- finds a sum of squares representation of f^3 findSOS by H. Peyrl and P. A. Parrilo 2007 #candidate points: 190 #points (even and within box of polynomial degrees): 43 #points in subspace of exponent-points: 43 #points inside Newton polytope: 19 Solving SOS feasibility problem... simpleSDP by H. Peyrl and J. Loefberg 2007 #It: b'y dy'Hdy mu alpha 1: | 3.448298 | 1868.12 0.1 0.0441942 2: | 1.482788 | 143.611 0.1 0.125 3: | 0.544693 | 18.8695 0.1 0.5 4: | 0.605315 | 5.84045 0.1 1. 5: | 0.731967 | 4.20856 0.1 1. 6: | 0.807068 | 3.29286 0.1 1. 7: | 0.877360 | 2.40634 0.1 1. 8: | 0.974919 | 1.26504 0.1 1. 9: | 1.059378 | 0.343443 0.1 1. 10: | -0.291699 | 1373.01 0.01 0.0883883 11: | -0.371348 | 5.80944 0.01 1. 12: | -0.360785 | 2.53961 0.01 1. 13: | -0.348581 | 1.31496 0.01 1. 14: | -0.339193 | 0.430715 0.01 1. 15: | -0.335381 | 0.0583206 0.01 1. 16: | -0.398360 | 906.386 0.001 0.0625 17: | -0.416144 | 52.1208 0.001 0.25 18: | -0.416544 | 0.629849 0.001 1. 19: | -0.416364 | 0.0734721 0.001 1. 20: | -0.420589 | 429.858 0.0001 0.0883883 21: | -0.420661 | 0.245141 0.0001 1. 22: | -0.420964 | 307.167 0.00001 0.0883883 23: | -0.421040 | 5.81918 0.00001 0.707107 24: | -0.421032 | 1.93267 0.00001 1. 25: | -0.421024 | 1.1705 0.00001 1. 26: | -0.421020 | 0.709412 0.00001 1. 27: | -0.421017 | 0.446044 0.00001 1. 28: | -0.421015 | 0.198415 0.00001 1. 29: | -0.421043 | 293.393 1.*10^-6 0.0883883 30: | -0.421048 | 7.3644 1.*10^-6 0.353553 31: | -0.421049 | 0.276948 1.*10^-6 1. rounding step #-3 Rounding precision: -3 Time needed for projection: 0.001 Time needed for LDL decomposition: 0.008999 rounding step #-2 Rounding precision: -2 Time needed for projection: 0.001 Time needed for LDL decomposition: 0.011998 rounding step #-1 Rounding precision: -1 Time needed for projection: 0.001 Time needed for LDL decomposition: 0.009999 rounding step #0 Rounding precision: 0 Time needed for projection: 0.001 Time needed for LDL decomposition: 0.012998 rounding step #1 Rounding precision: 1 Time needed for projection: 0.002 Time needed for LDL decomposition: 0.012998 459 4 4 1071 4 2 1071 2 4 2 2 17 17 6 3 1071 4 5 oo8 = ({- ----*x y - ----*x y - ----*x y + x y - --, - --*x y - ----*x y + 3650 3796 3796 73 73 3796 -------------------------------------------------------------------------- 4 3 459 2 3 1071 2 1071 5 4 17 3 6 3 4 459 3 2 x y - ----*x y - ----*x y, - ----*x y - --*x y + x y - ----*x y - 3650 3796 3796 73 3650 -------------------------------------------------------------------------- 1071 2 65670975 5 4 8569925 3 6 3 2 65670975 2 ----*x*y , - ---------*x y + --------*x y + x y - ---------*x*y , 3796 137237294 68618647 137237294 -------------------------------------------------------------------------- 8569925 6 3 65670975 4 5 2 3 65670975 2 4 4 --------*x y - ---------*x y + x y - ---------*x y, x y - 68618647 137237294 137237294 -------------------------------------------------------------------------- 65670975 4 2 65670975 2 4 8569925 175 5 3 175 3 5 3 3 ---------*x y - ---------*x y + --------, - ---*x y - ---*x y + x y - 137237294 137237294 68618647 629 629 -------------------------------------------------------------------------- 175 4 2 421805182124 2 4 80070895463 2 4 1201063431945 ---*x*y, x y - -------------*x y - -------------, x y - --------------, 629 9238122086595 1231749611546 17632633808942 -------------------------------------------------------------------------- 80070895463 6 3 421805182124 4 5 2 421805182124 5 4 - -------------*x y - -------------*x y + x y, - -------------*x y - 1231749611546 9238122086595 9238122086595 -------------------------------------------------------------------------- 80070895463 3 6 2 5 4 1201063431945 3 6 -------------*x y + x*y , x y - --------------*x y , - 1231749611546 17632633808942 -------------------------------------------------------------------------- 1201063431945 6 3 4 5 21157 5 3 21157 3 5 --------------*x y + x y , - ------*x y - ------*x y + x*y, - 17632633808942 107159 107159 -------------------------------------------------------------------------- 21157 5 3 3 5 6 3 5 3 3 6 146 146 146 4036391 4036391 -----*x y + x y , x y , 1, x y , x y }, {---, ---, ---, -------, -------, 86002 17 17 17 1186250 1186250 -------------------------------------------------------------------------- 4036391 74 1847624417319 431999528319079 1847624417319 1847624417319 -------, --, -------------, ---------------, -------------, -------------, 1186250 25 1971413728310 461906104329750 1971413728310 1971413728310 -------------------------------------------------------------------------- 431999528319079 431999528319079 8243 1032024 16675964223443 ---------------, ---------------, -----, -------, --------------, 461906104329750 461906104329750 10693 1393067 35265267617884 -------------------------------------------------------------------------- 16675964223443 389070 16675964223443 --------------, ------, --------------}) 35265267617884 559013 35265267617884 oo8 : Sequence ii9 : getSOS(f) ---- checks that f can not be written as a sum of squares. findSOS by H. Peyrl and P. A. Parrilo 2007 #candidate points: 28 #points (even and within box of polynomial degrees): 8 #points in subspace of exponent-points: 8 #points inside Newton polytope: 4 Solving SOS feasibility problem... simpleSDP by H. Peyrl and J. Loefberg 2007 #It: b'y dy'Hdy mu alpha 1: | 1.114189 | 0.0202703 0.1 1. 2: | 1.011565 | 104.291 0.01 0.0883883 3: | 1.009955 | 0.0193789 0.01 1. 4: | 1.002089 | 79.9169 0.001 0.0883883 5: | 1.000485 | 1.17959 0.001 0.707107 6: | 1.000735 | 0.266238 0.001 1. 7: | 1.000152 | 40.2912 0.0001 0.125 8: | 1.000073 | 0.268009 0.0001 1. 9: | 1.000015 | 39.9537 0.00001 0.125 10: | 1.000007 | 0.287906 0.00001 1. 11: | 1.000002 | 37.467 1.*10^-6 0.125 12: | 1.000001 | 0.452289 1.*10^-6 1. 13: | 1.000001 | 0.204565 1.*10^-6 1. rounding step #-3 Rounding precision: -3 Time needed for projection: 0. Time needed for LDL decomposition: 0.001 rounding step #-2 Rounding precision: -2 Time needed for projection: 0.001 Time needed for LDL decomposition: 0.001 ..... rounding step #51 Rounding precision: 51 Time needed for projection: 0. Time needed for LDL decomposition: 0.001 .Macaulay2/code/SOS/getSOS.m2:24:23:(1):[0]: Gram Matrix is not positive semidefinite stdio:7:1:(1):[0]: --back trace--