{"id":573,"date":"2013-04-15T12:00:34","date_gmt":"2013-04-15T17:00:34","guid":{"rendered":"https:\/\/blogs.mathworks.com\/cleve\/?p=573"},"modified":"2013-05-02T09:56:58","modified_gmt":"2013-05-02T14:56:58","slug":"wilkinsons-matrices-2","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/cleve\/2013\/04\/15\/wilkinsons-matrices-2\/","title":{"rendered":"Wilkinson&#8217;s Matrices"},"content":{"rendered":"\r\n\r\n<div class=\"content\"><!--introduction--><p>One of Jim Wilkinson's outstanding skills was his ability to pick great examples.  A previous post featured his signature polynomial. This post features his signature matrices.<\/p><!--\/introduction--><h3>Contents<\/h3><div><ul><li><a href=\"#b02f0d32-cc5b-4caf-84bf-004916d6943f\">Tridiagonal Matrices<\/a><\/li><li><a href=\"#05fe0fd2-337c-432e-8f89-360e8c60182c\">Wilkinson's Matrices<\/a><\/li><li><a href=\"#4df1ab29-2238-4cca-9e0e-25c85c1b77f6\">Eigenvalues<\/a><\/li><li><a href=\"#fc289bd0-3d97-4a48-b025-330addca7510\">Properties<\/a><\/li><li><a href=\"#7439e6e5-1366-4c86-b6e7-3ad2301c54d6\">Remarkable Closeness<\/a><\/li><li><a href=\"#9ee7bdf1-0660-46e0-b1e1-3b0f78aedb53\">Eigenvectors<\/a><\/li><li><a href=\"#eaf005b7-1853-4ac2-952c-ae3c216e2803\">Perturbation Theory<\/a><\/li><li><a href=\"#09aed841-3408-482a-806e-6ff493b1d1cf\">Acknowledgement<\/a><\/li><li><a href=\"#df8cc255-c9a2-498e-b1f0-306d00a068e4\">Reference<\/a><\/li><\/ul><\/div><h4>Tridiagonal Matrices<a name=\"b02f0d32-cc5b-4caf-84bf-004916d6943f\"><\/a><\/h4><p>When working with eigenvalues, we always have to be concerned about multiplicity and separation.  Multiple eigenvalues and clusters of close eigenvalues require careful attention.  Symmetric tridiagonal matrices are easier to handle than the general case.  If none of the off diagonal elements are zero, there are no multiple eigenvalues.  But if none of the off diagonal elements are \"small\", can we conclude that the eigenvalues are reasonably well separated?  One of Wilkinson's matrices answers this question with a resounding \"no\".<\/p><h4>Wilkinson's Matrices<a name=\"05fe0fd2-337c-432e-8f89-360e8c60182c\"><\/a><\/h4><p>Wilkinson introduces the matrices $W_{2n+1}^-$ and $W_{2n+1}^+$ on page 308 of <i>The Algebraic Eigenvalue Problem<\/i> and then employs them throughout the rest of the book.  These matrices are the families of symmetric tridiagonal matrices generated by the following MATLAB function. ($W_n^+$ is also generated by the function <tt>wilkinson<\/tt> in <tt>toolbox\\matlab\\elmat<\/tt> that has been part of MATLAB for many years.)<\/p><pre class=\"codeinput\">type <span class=\"string\">generate_wilkinson_matrices<\/span>\r\n<\/pre><pre class=\"codeoutput\">\r\nfunction [Wm,Wp] = generate_wilkinson_matrices(n)\r\n\r\n   D = diag(ones(2*n,1),1);\r\n   Wm = diag(-n:n) + D + D';\r\n   Wp = abs(diag(-n:n)) + D + D';\r\n\r\n<\/pre><p>Here, for example, is $n = 3, 2n+1 = 7$.<\/p><pre class=\"codeinput\">format <span class=\"string\">compact<\/span>\r\n[Wm,Wp] = generate_wilkinson_matrices(3)\r\n<\/pre><pre class=\"codeoutput\">Wm =\r\n    -3     1     0     0     0     0     0\r\n     1    -2     1     0     0     0     0\r\n     0     1    -1     1     0     0     0\r\n     0     0     1     0     1     0     0\r\n     0     0     0     1     1     1     0\r\n     0     0     0     0     1     2     1\r\n     0     0     0     0     0     1     3\r\nWp =\r\n     3     1     0     0     0     0     0\r\n     1     2     1     0     0     0     0\r\n     0     1     1     1     0     0     0\r\n     0     0     1     0     1     0     0\r\n     0     0     0     1     1     1     0\r\n     0     0     0     0     1     2     1\r\n     0     0     0     0     0     1     3\r\n<\/pre><h4>Eigenvalues<a name=\"4df1ab29-2238-4cca-9e0e-25c85c1b77f6\"><\/a><\/h4><p>Wilkinson usually has $2n+1 = 21$. Here are the eigenvalues of $W_{21}^-$ and $W_{21}^+$.<\/p><pre class=\"codeinput\">format <span class=\"string\">long<\/span>\r\n[Wm,Wp] = generate_wilkinson_matrices(10);\r\ndisp(<span class=\"string\">'       eig(Wm(21))         eig(Wp(21))'<\/span>)\r\ndisp(flipud([eig(Wm) eig(Wp)]))\r\n<\/pre><pre class=\"codeoutput\">       eig(Wm(21))         eig(Wp(21))\r\n  10.746194182903357  10.746194182903393\r\n   9.210678647333035  10.746194182903322\r\n   8.038941119306445   9.210678647361332\r\n   7.003952002665359   9.210678647304919\r\n   6.000225680185169   8.038941122829023\r\n   5.000008158672943   8.038941115814275\r\n   4.000000205070442   7.003952209528674\r\n   3.000000003808129   7.003951798616375\r\n   2.000000000054486   6.000234031584166\r\n   1.000000000000619   6.000217522257097\r\n   0.000000000000003   5.000244425001915\r\n  -1.000000000000618   4.999782477742903\r\n  -2.000000000054490   4.004354023440857\r\n  -3.000000003808127   3.996048201383625\r\n  -4.000000205070437   3.043099292578824\r\n  -5.000008158672947   2.961058884185726\r\n  -6.000225680185168   2.130209219362506\r\n  -7.003952002665363   1.789321352695084\r\n  -8.038941119306442   0.947534367529292\r\n  -9.210678647333047   0.253805817096678\r\n -10.746194182903357  -1.125441522119985\r\n<\/pre><h4>Properties<a name=\"fc289bd0-3d97-4a48-b025-330addca7510\"><\/a><\/h4><p>Here are some important properties of these eigenvalues.<\/p><div><ul><li>All the eigenvalues of $W_{2n+1}^-$, except the middle one,   occur in $\\pm$ pairs.<\/li><\/ul><\/div><div><ul><li>$W_{2n+1}^-$ is singular.  The middle eigenvalue is zero.<\/li><\/ul><\/div><div><ul><li>All the eigenvalues of $W_{2n+1}^+$, but one, are positive.<\/li><\/ul><\/div><div><ul><li>The positive eigenvalues of $W_{2n+1}^+$ occur roughly in pairs, with a   positive eigenvalue of $W_{2n+1}^-$ approximately in the center of   each pair.<\/li><\/ul><\/div><div><ul><li>The larger pairs of eigenvalues of $W_{2n+1}^+$ are incredibly close   together.<\/li><\/ul><\/div><h4>Remarkable Closeness<a name=\"7439e6e5-1366-4c86-b6e7-3ad2301c54d6\"><\/a><\/h4><p>All of the off diagonal elements in $W_{21}^+$ equal one, so none are \"small\", and yet the first two eigenvalues of $W_{21}^+$ agree to fourteen significant figures.  Wilkinson calls this \"remarkable\" and \"pathological\". The relative distance is<\/p><pre class=\"codeinput\">format <span class=\"string\">short<\/span> <span class=\"string\">e<\/span>\r\ne = flipud(eig(Wp));\r\ndelta = (e(1)-e(2))\/e(1)\r\n<\/pre><pre class=\"codeoutput\">delta =\r\n   6.6120e-15\r\n<\/pre><h4>Eigenvectors<a name=\"9ee7bdf1-0660-46e0-b1e1-3b0f78aedb53\"><\/a><\/h4><p>The behavior of the eigenvalues can be understood by looking at the eigenvectors.  Here is the vector $u_1$ associated with the first eigenvalue of $W_{21}^-$ and the vectors $v_1$ and $v_2$ associated with the leading pair of eigenvalues of $W_{21}^+$, Each vector has been normalized so that its first component is one.<\/p><pre class=\"codeinput\">[U,~] = eig(Wm);\r\n[V,~] = eig(Wp);\r\nu1 = flipud(U(:,end)\/U(end,end));\r\nv1 = flipud(V(:,end)\/V(end,end));\r\nv2 = flipud(V(:,end-1)\/V(end,end-1));\r\n\r\nformat <span class=\"string\">long<\/span>\r\n[u1 v1 v2]\r\n<\/pre><pre class=\"codeoutput\">ans =\r\n   1.000000000000000   1.000000000000000   1.000000000000000\r\n   0.746194182903358   0.746194182903392   0.746194182903321\r\n   0.302999941502167   0.302999941502254   0.302999941502075\r\n   0.085902493869951   0.085902493870164   0.085902493869724\r\n   0.018807481330335   0.018807481331049   0.018807481329571\r\n   0.003361464615150   0.003361464618322   0.003361464611748\r\n   0.000508147087273   0.000508147104788   0.000508147068490\r\n   0.000066594309069   0.000066594424058   0.000066594185758\r\n   0.000007705362252   0.000007706235465   0.000007704425844\r\n   0.000000798285440   0.000000805807738   0.000000790218740\r\n   0.000000074882661   0.000000147323229  -0.000000002800555\r\n   0.000000006418173   0.000000777356287  -0.000000820314052\r\n   0.000000000506441   0.000007428942095  -0.000007992139484\r\n   0.000000000037026   0.000064197613845  -0.000069080489810\r\n   0.000000000002522   0.000489858240829  -0.000527118748833\r\n   0.000000000000161   0.003240481200881  -0.003486964947267\r\n   0.000000000000010   0.018130575985481  -0.019509658947141\r\n   0.000000000000001   0.082810753074099  -0.089109664858083\r\n   0.000000000000000   0.292094585462557  -0.314312449184672\r\n   0.000000000000000   0.719337698380754  -0.774053354706959\r\n   0.000000000000000   0.964008718993031  -1.037335016061421\r\n<\/pre><p>And here is a plot of these three eigenvectors. The components of the first half of each of the three vectors decay rapidly and all three agree with each other to many decimal places.<\/p><pre class=\"codeinput\">n = 10;\r\nplot_wilkinson_eigenvectors\r\n<\/pre><img decoding=\"async\" vspace=\"5\" hspace=\"5\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/jhw_3_01.png\" alt=\"\"> <p>For a tridiagonal matrix $A$, the eigenvalue\/vector equation<\/p><p>$$ Ax = \\lambda x $$<\/p><p>is a three-term recurrence relation for the components of $x$. For the first half of each of the three vectors, we have the same positive elements on the diagonal of $A$, ones on the off diagonal, and nearly the same positive value of $\\lambda$, so the recurrence relation is nearly the same and leads to nearly the same rapidly decreasing components.<\/p><p>Once the recurrence relation gets half way through the vector and the components get small, this argument breaks down.  For $u_1$ the diagonal elements are negative and so the components of the vector remain small.  Perron-Frobenius implies that $v_1$ goes positive. Symmetry implies that $v_2$ goes negative to stay orthogonal to $v_1$.<\/p><h4>Perturbation Theory<a name=\"eaf005b7-1853-4ac2-952c-ae3c216e2803\"><\/a><\/h4><p>A closer examination of the elements of $v_1$ reveals that they decay exponentially until they reach a minimum half way through the vector at index $n+1$. It is possible to estimate <i>a priori<\/i> that this minimum value is of order $O(1\/n!)$<\/p><pre class=\"codeinput\">format <span class=\"string\">short<\/span> <span class=\"string\">e<\/span>\r\nminimum = v1(n+1)\r\nestimate = 1\/factorial(n)\r\n<\/pre><pre class=\"codeoutput\">minimum =\r\n   1.4732e-07\r\nestimate =\r\n   2.7557e-07\r\n<\/pre><p>This information about the shape of the dominant eigenvector makes it possible to prove that the corresponding eigenvalue, $\\lambda_1$, is a double eigenvalue with separation of order the square of the minimum in the eigenvector, which is $O(1\/(n!)^2)$.<\/p><pre class=\"codeinput\">separation = e(1)-e(2)\r\nestimate = 1\/factorial(n)^2\r\n<\/pre><pre class=\"codeoutput\">separation =\r\n   7.1054e-14\r\nestimate =\r\n   7.5941e-14\r\n<\/pre><h4>Acknowledgement<a name=\"09aed841-3408-482a-806e-6ff493b1d1cf\"><\/a><\/h4><p>Thanks to Pete Stewart, University of Maryland, for some helpful email while I was working on this post.<\/p><h4>Reference<a name=\"df8cc255-c9a2-498e-b1f0-306d00a068e4\"><\/a><\/h4><p>J. H. Wilkinson, <i>The Algebraic Eigenvalue Problem<\/i>, Claredon Press, Oxford, 1965, 662 pp.<\/p><script language=\"JavaScript\"> <!-- \r\n    function grabCode_60ab68bb02d44e72ae4b4f85e9855c4a() {\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='60ab68bb02d44e72ae4b4f85e9855c4a ' + '##### ' + 'SOURCE BEGIN' + ' #####';\r\n        t2='##### ' + 'SOURCE END' + ' #####' + ' 60ab68bb02d44e72ae4b4f85e9855c4a';\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_60ab68bb02d44e72ae4b4f85e9855c4a()\"><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\n60ab68bb02d44e72ae4b4f85e9855c4a ##### SOURCE BEGIN #####\r\n%% Wilkinson's Matrices\r\n% One of Jim Wilkinson's outstanding skills was his ability to pick\r\n% great examples.  A previous post featured his signature polynomial.\r\n% This post features his signature matrices. \r\n\r\n%% Tridiagonal Matrices\r\n% When working with eigenvalues, we always have to be concerned about\r\n% multiplicity and separation.  Multiple eigenvalues and clusters of close\r\n% eigenvalues require careful attention.  Symmetric tridiagonal matrices\r\n% are easier to handle than the general case.  If none of the off diagonal\r\n% elements are zero, there are no multiple eigenvalues.  But if none of the\r\n% off diagonal elements are \"small\", can we conclude that the eigenvalues\r\n% are reasonably well separated?  One of Wilkinson's matrices answers this\r\n% question with a resounding \"no\".\r\n\r\n%% Wilkinson's Matrices\r\n% Wilkinson introduces the matrices $W_{2n+1}^-$ and $W_{2n+1}^+$ on page 308\r\n% of _The Algebraic Eigenvalue Problem_ and then employs them throughout the\r\n% rest of the book.  These matrices are the families of symmetric\r\n% tridiagonal matrices generated by the following MATLAB function.\r\n% ($W_n^+$ is also generated by the function |wilkinson| in\r\n% |toolbox\\matlab\\elmat| that has been part of MATLAB for many years.)\r\n\r\ntype generate_wilkinson_matrices\r\n\r\n%%\r\n% Here, for example, is $n = 3, 2n+1 = 7$.\r\n\r\nformat compact\r\n[Wm,Wp] = generate_wilkinson_matrices(3)\r\n\r\n%% Eigenvalues\r\n% Wilkinson usually has $2n+1 = 21$.\r\n% Here are the eigenvalues of $W_{21}^-$ and $W_{21}^+$.\r\n\r\nformat long\r\n[Wm,Wp] = generate_wilkinson_matrices(10);\r\ndisp('       eig(Wm(21))         eig(Wp(21))')\r\ndisp(flipud([eig(Wm) eig(Wp)]))\r\n\r\n%% Properties\r\n% Here are some important properties of these eigenvalues.\r\n%\r\n% * All the eigenvalues of $W_{2n+1}^-$, except the middle one,\r\n%   occur in $\\pm$ pairs.\r\n%\r\n% * $W_{2n+1}^-$ is singular.  The middle eigenvalue is zero.\r\n% \r\n% * All the eigenvalues of $W_{2n+1}^+$, but one, are positive.\r\n%\r\n% * The positive eigenvalues of $W_{2n+1}^+$ occur roughly in pairs, with a \r\n%   positive eigenvalue of $W_{2n+1}^-$ approximately in the center of\r\n%   each pair.\r\n%\r\n% * The larger pairs of eigenvalues of $W_{2n+1}^+$ are incredibly close\r\n%   together.\r\n\r\n%% Remarkable Closeness\r\n% All of the off diagonal elements in $W_{21}^+$ equal one, so none are \"small\",\r\n% and yet the first two eigenvalues of $W_{21}^+$ agree to fourteen\r\n% significant figures.  Wilkinson calls this \"remarkable\" and \"pathological\".\r\n% The relative distance is\r\n\r\nformat short e\r\ne = flipud(eig(Wp));\r\ndelta = (e(1)-e(2))\/e(1)\r\n\r\n%% Eigenvectors\r\n% The behavior of the eigenvalues can be understood by looking at the\r\n% eigenvectors.  Here is the vector $u_1$ associated with the first\r\n% eigenvalue of $W_{21}^-$ and the vectors $v_1$ and $v_2$ associated with\r\n% the leading pair of eigenvalues of $W_{21}^+$, Each vector has been\r\n% normalized so that its first component is one.\r\n \r\n[U,~] = eig(Wm);\r\n[V,~] = eig(Wp);\r\nu1 = flipud(U(:,end)\/U(end,end));\r\nv1 = flipud(V(:,end)\/V(end,end));\r\nv2 = flipud(V(:,end-1)\/V(end,end-1));\r\n\r\nformat long\r\n[u1 v1 v2]\r\n\r\n%%\r\n% And here is a plot of these three eigenvectors.\r\n% The components of the first half of each of the three vectors decay\r\n% rapidly and all three agree with each other to many decimal places.\r\n\r\nn = 10;\r\nplot_wilkinson_eigenvectors\r\n\r\n%%\r\n% For a tridiagonal matrix $A$, the eigenvalue\/vector equation\r\n%\r\n% $$ Ax = \\lambda x $$\r\n%\r\n% is a three-term recurrence relation for the components of $x$.\r\n% For the first half of each of the three vectors, we have the same\r\n% positive elements on the diagonal of $A$, ones on the off diagonal,\r\n% and nearly the same positive value of $\\lambda$, so the recurrence\r\n% relation is nearly the same and leads to nearly the same\r\n% rapidly decreasing components.  \r\n%\r\n% Once the recurrence relation gets half way through the vector and\r\n% the components get small, this argument breaks down.  For $u_1$ the\r\n% diagonal elements are negative and so the components of the vector\r\n% remain small.  Perron-Frobenius implies that $v_1$ goes positive.\r\n% Symmetry implies that $v_2$ goes negative to stay orthogonal to $v_1$.\r\n%\r\n\r\n%% Perturbation Theory\r\n% A closer examination of the elements of $v_1$ reveals that they decay\r\n% exponentially until they reach a minimum half way through the vector\r\n% at index $n+1$.\r\n% It is possible to estimate _a priori_ that this minimum value is of\r\n% order $O(1\/n!)$\r\n\r\nformat short e\r\nminimum = v1(n+1)\r\nestimate = 1\/factorial(n)\r\n\r\n%%\r\n% This information about the shape of the dominant eigenvector makes\r\n% it possible to prove that the corresponding eigenvalue, $\\lambda_1$, is\r\n% a double eigenvalue with separation of order the square of the minimum\r\n% in the eigenvector, which is $O(1\/(n!)^2)$.\r\n\r\nseparation = e(1)-e(2)\r\nestimate = 1\/factorial(n)^2\r\n\r\n\r\n%% Acknowledgement\r\n% Thanks to Pete Stewart, University of Maryland, for some helpful email\r\n% while I was working on this post.\r\n\r\n%% Reference\r\n% J. H. Wilkinson, _The Algebraic Eigenvalue Problem_, Claredon Press,\r\n% Oxford, 1965, 662 pp.\r\n\r\n##### SOURCE END ##### 60ab68bb02d44e72ae4b4f85e9855c4a\r\n-->","protected":false},"excerpt":{"rendered":"<div class=\"overview-image\"><img decoding=\"async\"  class=\"img-responsive\" src=\"https:\/\/blogs.mathworks.com\/images\/cleve\/jhw_3_01.png\" onError=\"this.style.display ='none';\" \/><\/div><!--introduction--><p>One of Jim Wilkinson's outstanding skills was his ability to pick great examples.  A previous post featured his signature polynomial. This post features his signature matrices.... <a class=\"read-more\" href=\"https:\/\/blogs.mathworks.com\/cleve\/2013\/04\/15\/wilkinsons-matrices-2\/\">read more >><\/a><\/p>","protected":false},"author":78,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[6],"tags":[],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/573"}],"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=573"}],"version-history":[{"count":13,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/573\/revisions"}],"predecessor-version":[{"id":622,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/posts\/573\/revisions\/622"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/media?parent=573"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/categories?post=573"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/cleve\/wp-json\/wp\/v2\/tags?post=573"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}