{"id":538,"date":"2013-03-04T12:00:17","date_gmt":"2013-03-04T17:00:17","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=538"},"modified":"2013-05-02T09:59:19","modified_gmt":"2013-05-02T14:59:19","slug":"wilkinsons-polynomials","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2013\/03\/04\/wilkinsons-polynomials\/","title":{"rendered":"Wilkinson&#8217;s Polynomials"},"content":{"rendered":"<!DOCTYPE html\r\n  PUBLIC \"-\/\/W3C\/\/DTD HTML 4.01 Transitional\/\/EN\">\r\n<style type=\"text\/css\">\r\n\r\nh1 { font-size:18pt; }\r\nh2.titlebg { font-size:13pt; }\r\nh3 { color:#4A4F55; padding:0px; margin:5px 0px 5px; font-family:Arial, Helvetica, sans-serif; font-size:11pt; font-weight:bold; line-height:140%; border-bottom:1px solid #d6d4d4; display:block; }\r\nh4 { color:#4A4F55; padding:0px; margin:0px 0px 5px; font-family:Arial, Helvetica, sans-serif; font-size:10pt; font-weight:bold; line-height:140%; border-bottom:1px solid #d6d4d4; display:block; }\r\n   \r\np { padding:0px; margin:0px 0px 20px; }\r\nimg { padding:0px; margin:0px 0px 20px; border:none; }\r\np img, pre img, tt img, li img { margin-bottom:0px; } \r\n\r\nul { padding:0px; margin:0px 0px 20px 23px; list-style:square; }\r\nul li { padding:0px; margin:0px 0px 7px 0px; background:none; }\r\nul li ul { padding:5px 0px 0px; margin:0px 0px 7px 23px; }\r\nul li ol li { list-style:decimal; }\r\nol { padding:0px; margin:0px 0px 20px 0px; list-style:decimal; }\r\nol li { padding:0px; margin:0px 0px 7px 23px; list-style-type:decimal; }\r\nol li ol { padding:5px 0px 0px; margin:0px 0px 7px 0px; }\r\nol li ol li { list-style-type:lower-alpha; }\r\nol li ul { padding-top:7px; }\r\nol li ul li { list-style:square; }\r\n\r\npre, tt, code { font-size:12px; }\r\npre { margin:0px 0px 20px; }\r\npre.error { color:red; }\r\npre.codeinput { padding:10px; border:1px solid #d3d3d3; background:#f7f7f7; }\r\npre.codeoutput { padding:10px 11px; margin:0px 0px 20px; color:#4c4c4c; }\r\n\r\n@media print { pre.codeinput, pre.codeoutput { word-wrap:break-word; width:100%; } }\r\n\r\nspan.keyword { color:#0000FF }\r\nspan.comment { color:#228B22 }\r\nspan.string { color:#A020F0 }\r\nspan.untermstring { color:#B20000 }\r\nspan.syscmd { color:#B28C00 }\r\n\r\n.footer { width:auto; padding:10px 0px; margin:25px 0px 0px; border-top:1px dotted #878787; font-size:0.8em; line-height:140%; font-style:italic; color:#878787; text-align:left; float:none; }\r\n.footer p { margin:0px; }\r\n\r\n  <\/style><div class=\"content\"><!--introduction--><p>Wilkinson's polynomials are a family of polynmials with deceptively sensitive roots.<\/p><!--\/introduction--><h3>Contents<\/h3><div><ul><li><a href=\"#c06781ed-5c04-4bc2-a995-a97a9375e1a8\">$P_n$<\/a><\/li><li><a href=\"#0ef11e99-e8c3-4bcf-9cbc-1060cf7e1ba7\">$P_{20}$<\/a><\/li><li><a href=\"#08c9a106-718c-4073-9824-4cfe53418dd5\">Coefficients<\/a><\/li><li><a href=\"#ae75acb0-6dc5-4c82-923b-daa852d4ec51\">Symbolic form<\/a><\/li><li><a href=\"#ebaccbdf-d4f0-4adc-bd87-05436ba95bfc\">Double precision<\/a><\/li><li><a href=\"#88810d87-ef68-452f-8fb5-aecb7e711501\">Double precision roots<\/a><\/li><li><a href=\"#4e933ac9-12f6-4685-af5a-d59152175246\">Wilkinson's perturbation<\/a><\/li><li><a href=\"#aeb2589c-c6bb-44c1-8ea7-73c838dbf237\">Sensitivity<\/a><\/li><li><a href=\"#02259bc3-4e57-4579-9790-3f23b2535e68\">Note added March 10, 2013.<\/a><\/li><\/ul><\/div><h4>$P_n$<a name=\"c06781ed-5c04-4bc2-a995-a97a9375e1a8\"><\/a><\/h4><p>The Wilkinson polynomial of degree $n$ has as roots the integers from $1$ to $n$.<\/p><p>$$ P_n(x) = \\prod_{k=1}^{n} (x-k) $$<\/p><h4>$P_{20}$<a name=\"0ef11e99-e8c3-4bcf-9cbc-1060cf7e1ba7\"><\/a><\/h4><p>The usual value of $n$ is 20.<\/p><pre class=\"codeinput\">format <span class=\"string\">compact<\/span>\r\nn = 20\r\nsyms <span class=\"string\">x<\/span>\r\nP20 = prod(x-(1:n))\r\n<\/pre><pre class=\"codeoutput\">n =\r\n    20\r\nP20 =\r\n(x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)*(x - 6)*(x - 7)*(x - 8)*(x - 9)*(x - 10)*(x - 11)*(x - 12)*(x - 13)*(x - 14)*(x - 15)*(x - 16)*(x - 17)*(x - 18)*(x - 19)*(x - 20)\r\n<\/pre><p>Expressed in this form, the polynomial is well behaved.<\/p><h4>Coefficients<a name=\"08c9a106-718c-4073-9824-4cfe53418dd5\"><\/a><\/h4><p>It gets interesting when $P_n(x)$ is expanded in the monomial basis, $\\{x^j\\}$.<\/p><pre class=\"codeinput\">P = expand(P20)\r\n<\/pre><pre class=\"codeoutput\">P =\r\nx^20 - 210*x^19 + 20615*x^18 - 1256850*x^17 + 53327946*x^16 - 1672280820*x^15 + 40171771630*x^14 - 756111184500*x^13 + 11310276995381*x^12 - 135585182899530*x^11 + 1307535010540395*x^10 - 10142299865511450*x^9 + 63030812099294896*x^8 - 311333643161390640*x^7 + 1206647803780373360*x^6 - 3599979517947607200*x^5 + 8037811822645051776*x^4 - 12870931245150988800*x^3 + 13803759753640704000*x^2 - 8752948036761600000*x + 2432902008176640000\r\n<\/pre><p>The coefficients are huge.  The constant term is $20!$, and that's not even the largest coefficient.<\/p><pre class=\"codeinput\">C = flipud(coeffs(P)')\r\n<\/pre><pre class=\"codeoutput\">C =\r\n                     1\r\n                  -210\r\n                 20615\r\n              -1256850\r\n              53327946\r\n           -1672280820\r\n           40171771630\r\n         -756111184500\r\n        11310276995381\r\n      -135585182899530\r\n      1307535010540395\r\n    -10142299865511450\r\n     63030812099294896\r\n   -311333643161390640\r\n   1206647803780373360\r\n  -3599979517947607200\r\n   8037811822645051776\r\n -12870931245150988800\r\n  13803759753640704000\r\n  -8752948036761600000\r\n   2432902008176640000\r\n<\/pre><h4>Symbolic form<a name=\"ae75acb0-6dc5-4c82-923b-daa852d4ec51\"><\/a><\/h4><p>Let's have the Symbolic Toolbox generate LaTeX and cut and paste the result into the source for the blog.<\/p><pre class=\"codeinput\">L = latex(P);\r\n<span class=\"comment\">% edit(L)<\/span>\r\n<\/pre><p>$$ x^{20} - 210\\, x^{19} + 20615\\, x^{18} \\\\\r\n- 1256850\\, x^{17} + 53327946\\, x^{16} - 1672280820\\, x^{15} \\\\\r\n+ 40171771630\\, x^{14} - 756111184500\\, x^{13} + 11310276995381\\, x^{12} \\\\\r\n- 135585182899530\\, x^{11} + 1307535010540395\\, x^{10} - 10142299865511450\\, x^9 \\\\\r\n+ 63030812099294896\\, x^8 - 311333643161390640\\, x^7 + 1206647803780373360\\, x^6 \\\\\r\n- 3599979517947607200\\, x^5 + 8037811822645051776\\, x^4 - 12870931245150988800\\, x^3 \\\\\r\n+ 13803759753640704000\\, x^2 - 8752948036761600000\\, x + 2432902008176640000 $$<\/p><p>With the monomial basis the Wilkinson polynomial does not look so innocent. The spacing between the roots is tiny relative to these coefficients. But the toolbox can easily find the roots $1$ to $20$ with no error.<\/p><pre class=\"codeinput\">Z = sort(solve(P))'\r\n<\/pre><pre class=\"codeoutput\">Z =\r\n[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]\r\n<\/pre><h4>Double precision<a name=\"ebaccbdf-d4f0-4adc-bd87-05436ba95bfc\"><\/a><\/h4><p>Convert the symbolic form to double precision floating point.<\/p><pre class=\"codeinput\">format <span class=\"string\">long<\/span> <span class=\"string\">e<\/span>\r\np = sym2poly(P)';\r\nc = flipud(coeffs(poly2sym(p))')\r\n<\/pre><pre class=\"codeoutput\">c =\r\n                     1\r\n                  -210\r\n                 20615\r\n              -1256850\r\n              53327946\r\n           -1672280820\r\n           40171771630\r\n         -756111184500\r\n        11310276995381\r\n      -135585182899530\r\n      1307535010540395\r\n    -10142299865511450\r\n     63030812099294896\r\n   -311333643161390656\r\n   1206647803780373248\r\n  -3599979517947607040\r\n   8037811822645051392\r\n -12870931245150988288\r\n  13803759753640704000\r\n  -8752948036761600000\r\n   2432902008176640000\r\n<\/pre><p>Five of the coefficients cannot be represented in double precision format. They have been perturbed.<\/p><pre class=\"codeinput\">delta = C-c\r\n<\/pre><pre class=\"codeoutput\">delta =\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n    0\r\n   16\r\n  112\r\n -160\r\n  384\r\n -512\r\n    0\r\n    0\r\n    0\r\n<\/pre><h4>Double precision roots<a name=\"88810d87-ef68-452f-8fb5-aecb7e711501\"><\/a><\/h4><p>The roots of the polynomial with the double precision coefficients are no longer <tt>1:20<\/tt>. The largest perturbations occur to the roots at <tt>16<\/tt> and <tt>17<\/tt>.<\/p><pre class=\"codeinput\">z = sort(roots(p));\r\nfmt = <span class=\"string\">'%25.16f\\n'<\/span>;\r\nfprintf(fmt,z)\r\n<\/pre><pre class=\"codeoutput\">       0.9999999999997972\r\n       2.0000000000475513\r\n       2.9999999973842115\r\n       4.0000000600349486\r\n       4.9999992200080809\r\n       6.0000064898423089\r\n       6.9999645183068031\r\n       8.0001214537649652\r\n       8.9997977879058215\r\n       9.9997884273944386\r\n      11.0021078422257670\r\n      11.9944660326764140\r\n      13.0076639325972590\r\n      13.9958171375909190\r\n      14.9955431043450050\r\n      16.0114122070902880\r\n      16.9888815075391190\r\n      18.0060272186900080\r\n      18.9981637365860050\r\n      20.0002393259700360\r\n<\/pre><h4>Wilkinson's perturbation<a name=\"4e933ac9-12f6-4685-af5a-d59152175246\"><\/a><\/h4><p>Wilkinson made a different perturbation, a deliberate roundoff error on his machine to the second coefficient, the $-210$.  He changed this coefficient by $2^{-23}$ and discovered that several of the roots were driven into the complex plane.<\/p><p>I am not sure $^\\dagger$ about the sign of Wilkinson's perturbation, so let's do both. Here is a movie, an animated GIF, of the root locus in the complex plane produced by perturbations like his.  It shows the trajectories of the roots from $9$ to $20$ of<\/p><p>$$ P_{20}(x) - \\alpha x^{19} $$<\/p><p>as we vary $\\alpha$ over the range<\/p><p>$$ \\alpha = \\pm 2^{-k}, k = 23, ..., 36 $$ The roots $1$ to $8$ stay real for perturbations in this range. Wilkinson's result is at the end of either the red or the black trajectories.<\/p><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/P20_movie.gif\" alt=\"\"> <\/p><h4>Sensitivity<a name=\"aeb2589c-c6bb-44c1-8ea7-73c838dbf237\"><\/a><\/h4><p>With a little calculus, we can get an analytic expression for the sensitivity of the roots with Wilkinson's perturbation. Regard each root $x$ as function of $\\alpha$ and differentiate the following equation with respect to $\\alpha$.<\/p><p>$$ P_{20}(x) - \\alpha x^{19} = 0 $$<\/p><p>At $\\alpha = 0$, we have<\/p><p>$$ \\frac{dx}{d\\alpha} = \\frac{x^{19}}{P_{20}'(x)} $$<\/p><p>This is easy to evaluate.<\/p><pre class=\"codeinput\">pprime = sym2poly(diff(P20));\r\nxdot = zeros(n,1);\r\n<span class=\"keyword\">for<\/span> k = 1:n\r\n   xdot(k) = k^19\/polyval(pprime,k);\r\n<span class=\"keyword\">end<\/span>\r\nformat <span class=\"string\">short<\/span> <span class=\"string\">e<\/span>\r\nxdot\r\n<\/pre><pre class=\"codeoutput\">xdot =\r\n  -8.2206e-18\r\n   8.1890e-11\r\n  -1.6338e-06\r\n   2.1896e-03\r\n  -6.0774e-01\r\n   5.8248e+01\r\n  -2.5424e+03\r\n   5.9698e+04\r\n  -8.3916e+05\r\n   7.5994e+06\r\n  -4.6378e+07\r\n   1.9894e+08\r\n  -6.0434e+08\r\n   1.3336e+09\r\n  -2.1150e+09\r\n   2.4094e+09\r\n  -1.9035e+09\r\n   9.9571e+08\r\n  -3.0901e+08\r\n   4.3100e+07\r\n<\/pre><p>The sensitivities vary over 27 orders of magnitude, with the largest values again at $16$ and $17$.  Here is a log plot.<\/p><p><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/P20_prime.png\" alt=\"\"> <\/p><p>This is a different perturbation than the one from symbolic to double, but the qualitative effect is the same.<\/p><h4>Note added March 10, 2013.<a name=\"02259bc3-4e57-4579-9790-3f23b2535e68\"><\/a><\/h4><p>$^\\dagger$ I am back in the office where I have access to Wilkinson's <i>The Algebraic Eigenvalue Problem<\/i>.  This polynomial is discussed, among other places, on pages 417 and 418.  The perturbation he makes is negative, so the coefficient of $x^{19}$ becomes $-210 - 2^{-23}$ and the resulting roots are at the ends of our red trajectories.<\/p><script language=\"JavaScript\"> <!-- \r\n    function grabCode_4443b6643f054b068392bd39a37a23b9() {\r\n        \/\/ Remember the title so we can use it in the new page\r\n        title = document.title;\r\n\r\n        \/\/ Break up these strings so that their presence\r\n        \/\/ in the Javascript doesn't mess up the search for\r\n        \/\/ the MATLAB code.\r\n        t1='4443b6643f054b068392bd39a37a23b9 ' + '##### ' + 'SOURCE BEGIN' + ' #####';\r\n        t2='##### ' + 'SOURCE END' + ' #####' + ' 4443b6643f054b068392bd39a37a23b9';\r\n    \r\n        b=document.getElementsByTagName('body')[0];\r\n        i1=b.innerHTML.indexOf(t1)+t1.length;\r\n        i2=b.innerHTML.indexOf(t2);\r\n \r\n        code_string = b.innerHTML.substring(i1, i2);\r\n        code_string = code_string.replace(\/REPLACE_WITH_DASH_DASH\/g,'--');\r\n\r\n        \/\/ Use \/x3C\/g instead of the less-than character to avoid errors \r\n        \/\/ in the XML parser.\r\n        \/\/ Use '\\x26#60;' instead of '<' so that the XML parser\r\n        \/\/ doesn't go ahead and substitute the less-than character. \r\n        code_string = code_string.replace(\/\\x3C\/g, '\\x26#60;');\r\n\r\n        copyright = 'Copyright 2013 The MathWorks, Inc.';\r\n\r\n        w = window.open();\r\n        d = w.document;\r\n        d.write('<pre>\\n');\r\n        d.write(code_string);\r\n\r\n        \/\/ Add copyright line at the bottom if specified.\r\n        if (copyright.length > 0) {\r\n            d.writeln('');\r\n            d.writeln('%%');\r\n            if (copyright.length > 0) {\r\n                d.writeln('% _' + copyright + '_');\r\n            }\r\n        }\r\n\r\n        d.write('<\/pre>\\n');\r\n\r\n        d.title = title + ' (MATLAB code)';\r\n        d.close();\r\n    }   \r\n     --> <\/script><p style=\"text-align: right; font-size: xx-small; font-weight:lighter;   font-style: italic; color: gray\"><br><a href=\"javascript:grabCode_4443b6643f054b068392bd39a37a23b9()\"><span style=\"font-size: x-small;        font-style: italic;\">Get \r\n      the MATLAB code <noscript>(requires JavaScript)<\/noscript><\/span><\/a><br><br>\r\n      Published with MATLAB&reg; R2012b<br><\/p><p class=\"footer\"><br>\r\n      Published with MATLAB&reg; R2012b<br><\/p><\/div><!--\r\n4443b6643f054b068392bd39a37a23b9 ##### SOURCE BEGIN #####\r\n%% Wilkinson Polynomials\r\n% Wilkinson's polynomials are a family of polynmials with deceptively\r\n% sensitive roots.\r\n\r\n%% $P_n$\r\n% The Wilkinson polynomial of degree $n$ has as roots the integers from \r\n% $1$ to $n$. \r\n%\r\n% $$ P_n(x) = \\prod_{k=1}^{n} (x-k) $$\r\n%\r\n\r\n%% $P_{20}$\r\n% The usual value of $n$ is 20.\r\n\r\nformat compact\r\nn = 20\r\nsyms x\r\nP20 = prod(x-(1:n))\r\n\r\n%%\r\n% Expressed in this form, the polynomial is well behaved.\r\n\r\n%% Coefficients\r\n% It gets interesting when $P_n(x)$ is expanded in the monomial basis,\r\n% $\\{x^j\\}$.\r\n\r\nP = expand(P20)\r\n\r\n%%\r\n% The coefficients are huge.  The constant term is $20!$, and that's not even\r\n% the largest coefficient.\r\n\r\nC = flipud(coeffs(P)')\r\n\r\n%% Symbolic form\r\n% Let's have the Symbolic Toolbox generate LaTeX and cut and paste the result\r\n% into the source for the blog.\r\n\r\nL = latex(P);\r\n% edit(L)\r\n\r\n%%\r\n%\r\n% $$ x^{20} - 210\\, x^{19} + 20615\\, x^{18} \\\\\r\n% - 1256850\\, x^{17} + 53327946\\, x^{16} - 1672280820\\, x^{15} \\\\\r\n% + 40171771630\\, x^{14} - 756111184500\\, x^{13} + 11310276995381\\, x^{12} \\\\\r\n% - 135585182899530\\, x^{11} + 1307535010540395\\, x^{10} - 10142299865511450\\, x^9 \\\\\r\n% + 63030812099294896\\, x^8 - 311333643161390640\\, x^7 + 1206647803780373360\\, x^6 \\\\\r\n% - 3599979517947607200\\, x^5 + 8037811822645051776\\, x^4 - 12870931245150988800\\, x^3 \\\\\r\n% + 13803759753640704000\\, x^2 - 8752948036761600000\\, x + 2432902008176640000 $$\r\n\r\n%%\r\n% With the monomial basis the Wilkinson polynomial does not look so innocent.\r\n% The spacing between the roots is tiny relative to these coefficients.\r\n% But the toolbox can easily find the roots $1$ to $20$ with no error.\r\n\r\nZ = sort(solve(P))'\r\n\r\n%% Double precision\r\n% Convert the symbolic form to double precision floating point.\r\n\r\nformat long e\r\np = sym2poly(P)';\r\nc = flipud(coeffs(poly2sym(p))')\r\n\r\n%%\r\n% Five of the coefficients cannot be represented in double precision format.\r\n% They have been perturbed. \r\n\r\ndelta = C-c\r\n\r\n%% Double precision roots\r\n% The roots of the polynomial with the double precision coefficients\r\n% are no longer |1:20|.\r\n% The largest perturbations occur to the roots at |16| and |17|.\r\n\r\nz = sort(roots(p));\r\nfmt = '%25.16f\\n';\r\nfprintf(fmt,z)\r\n\r\n%% Wilkinson's perturbation\r\n% Wilkinson made a different perturbation, a deliberate roundoff\r\n% error on his machine to the second coefficient, the $-210$.  He changed\r\n% this coefficient by $2^{-23}$ and discovered that several of the roots\r\n% were driven into the complex plane.\r\n%\r\n% I am not sure $^\\dagger$\r\n% about the sign of Wilkinson's perturbation, so let's do both.\r\n% Here is a movie, an animated GIF, of the root locus in the complex plane\r\n% produced by perturbations like his.  It shows the trajectories of the roots\r\n% from $9$ to $20$ of\r\n%\r\n% $$ P_{20}(x) - \\alpha x^{19} $$\r\n%\r\n% as we vary $\\alpha$ over the range\r\n% \r\n% $$ \\alpha = \\pm 2^{-k}, k = 23, ..., 36 $$\r\n% The roots $1$ to $8$ stay real for perturbations in this range.\r\n% Wilkinson's result is at the end of either the red or the black trajectories.\r\n%\r\n% <<P20_movie.gif>>\r\n%\r\n\r\n%% Sensitivity\r\n% With a little calculus, we can get an analytic expression for the\r\n% sensitivity of the roots with Wilkinson's perturbation.\r\n% Regard each root $x$ as function of $\\alpha$ and differentiate the following\r\n% equation with respect to $\\alpha$.\r\n%\r\n% $$ P_{20}(x) - \\alpha x^{19} = 0 $$\r\n%\r\n% At $\\alpha = 0$, we have\r\n% \r\n% $$ \\frac{dx}{d\\alpha} = \\frac{x^{19}}{P_{20}'(x)} $$\r\n%\r\n% This is easy to evaluate.\r\n\r\npprime = sym2poly(diff(P20));\r\nxdot = zeros(n,1);\r\nfor k = 1:n\r\n   xdot(k) = k^19\/polyval(pprime,k);\r\nend \r\nformat short e\r\nxdot\r\n\r\n%%\r\n% The sensitivities vary over 27 orders of magnitude, with the largest\r\n% values again at $16$ and $17$.  Here is a log plot.\r\n%\r\n% <<P20_prime.png>>\r\n%\r\n\r\n%%\r\n% This is a different perturbation than the one from symbolic to double,\r\n% but the qualitative effect is the same.\r\n\r\n%% Note added March 10, 2013.\r\n% $^\\dagger$ I am back in the office where I have access to Wilkinson's\r\n% _The Algebraic Eigenvalue Problem_.  This polynomial is discussed,\r\n% among other places, on pages 417 and 418.  The perturbation he makes\r\n% is negative, so the coefficient of $x^{19}$ becomes $-210 - 2^{-23}$\r\n% and the resulting roots are at the ends of our red trajectories.\r\n\r\n##### SOURCE END ##### 4443b6643f054b068392bd39a37a23b9\r\n-->","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img decoding=\"async\"  class=\"img-responsive\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/P20_movie.gif\" onError=\"this.style.display ='none';\" \/><\/div><!--introduction--><p>Wilkinson's polynomials are a family of polynmials with deceptively sensitive roots.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2013\/03\/04\/wilkinsons-polynomials\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[7],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/538"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/users\/78"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/comments?post=538"}],"version-history":[{"count":15,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/538\/revisions"}],"predecessor-version":[{"id":625,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/538\/revisions\/625"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=538"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=538"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=538"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}