History of Condition numbers of the classical test matrices

 Title: Condition numbers of the classical test matrices-Definition: For a named classical test matrix $A_n$, this table gives the finite 2-norm-  condition number $\kappa_2(A_n)=\sigma_{\max}(A_n)/\sigma_{\min}(A_n)$ CITE{WikiCondition},-  where the singular values are taken in the Euclidean norm and $n$ is the order of-  the matrix.+Definition: For a named classical test matrix $A_n$ of order $n$, this table gives+  the 2-norm condition number $\kappa_2(A_n)=\sigma_{\max}(A_n)/\sigma_{\min}(A_n)$+  CITE{WikiCondition}, the ratio of the largest to the smallest singular value. Keywords: - condition number

       cauchy: Cauchy matrix       fiedler: Fiedler matrix-      kms: Kac-Murdock-Szego matrix+      kms: Kac-Murdock-Szegő matrix       minij: Minij matrix       grcar: Grcar matrix

     matrix CITE{WikiVandermonde}. The Cauchy matrix is $A_{ij}=1/(i+j)$, the default     scalar case of `gallery("cauchy",n)` CITE{MathWorksGallery}. The Fiedler matrix-    is $A_{ij}=|i-j|$. The Kac-Murdock-Szego matrix is $A_{ij}=2^{-|i-j|}$. The Minij+    is $A_{ij}=|i-j|$. The Kac-Murdock-Szegő matrix is $A_{ij}=2^{-|i-j|}$. The Minij     matrix is $A_{ij}=\min(i,j)$. The Grcar matrix has $-1$ on the subdiagonal and     $1$ on the diagonal and on the first three superdiagonals CITE{MathWorksGallery}-    CITE{TMTGallery}.-  comment-toolbox-families: The Frank, Clement, Moler, Parter, Ris, Lotkin and Riemann-    matrices use the definitions in the Test Matrix Toolbox source CITE{TMTGallery}.+    CITE{TMTGallery}. The second-difference matrix is tridiagonal, with $2$ on the+    diagonal and $-1$ on the first subdiagonal and first superdiagonal.+  comment-toolbox-families: The Frank, Wilkinson, Clement, Moler, Parter, Ris, Lotkin+    and Riemann matrices use the definitions in the Test Matrix Toolbox source CITE{TMTGallery}.     The Frank matrix is the unreflected upper Hessenberg form $A_{ij}=n-\max(i,j)+1$     for $j\geq i-1$, and $0$ otherwise. The Wilkinson matrix is symmetric tridiagonal

     and $A_{i,i+1}=i$. The Moler matrix has $A_{ii}=i$ and $A_{ij}=\min(i,j)-2$ for     $i\neq j$. The Parter matrix is $A_{ij}=1/(i-j+1/2)$. The Ris matrix, also called-    the Dingdong matrix, is $A_{ij}=(1/2)/(n-i-j+3/2)$. The Lotkin matrix is the Hilbert-    matrix with first row replaced by ones. The Riemann matrix is $B(2:n+1,2:n+1)$-    with $B_{ij}=i-1$ if $i\mid j$ and $-1$ otherwise.+    the Dingdong matrix, is $A_{ij}=(1/2)/(n-i-j+3/2)$. The Ris matrix is one half+    of the Parter matrix with its rows reversed, so the two have the same condition+    number at every order. The Lotkin matrix is the Hilbert matrix with first row+    replaced by ones. The Riemann matrix is the $n\times n$ matrix $A_{ij}=i$ if $(i+1)\mid(j+1)$+    and $-1$ otherwise.   comment-finite: Only finite condition numbers are entries. If a default matrix is-    singular, $\sigma_{\min}(A_n)=0$ and $\kappa_2(A_n)$ is not a real number stored-    by this table. This excludes the singular default Chow matrices for $n>1$ and-    singular orders of any other family.+    singular, $\sigma_{\min}(A_n)=0$ and $\kappa_2(A_n)$ is infinite rather than a+    real number, so there is no entry. This excludes the singular default Chow matrices+    for $n>1$ and singular orders of any other family.   comment-source: The MathWorks `gallery` documentation CITE{MathWorksGallery} states     the default parameter choices used here for the gallery families. The Test Matrix

   program-sage:     language: Sage-    code: 'import numberdb.sage as numberdb--      from sage.matrix.constructor import matrix+    code: 'from sage.matrix.constructor import matrix        from sage.rings.rational_field import QQ --      A = matrix(QQ, 3, 3, lambda i, j: QQ(1) / QQ(i + j + 1))+      from sage.rings.qqbar import AA+++      A = matrix(QQ, 3, 3, lambda i, j: QQ(1) / QQ((i + 1) + (j + 1) - 1))        B = A.transpose() * A -      B.eigenvalues()'+      ev = sorted(B.charpoly().roots(AA, multiplicities=False))++      (ev[-1] / ev[0]).sqrt().n(400)' Similar tables: - table: HREF{Cos_pi_times_x_at_rational_numbers}[$\cos(\pi x)$ for rational $x$]

     by exact determinant computation and omitted, because their 2-norm condition numbers     are infinite rather than finite real numbers. Before the entries were written,-    `agents/table-build/dry_run.py` computed all 351 entries; the Frank determinants-    were checked to be $1$; the Redheffer determinants were checked against the Mertens-    function; the Clement characteristic polynomials were checked against the eigenvalues-    $\pm(n-1),\pm(n-3),\ldots$; the second-difference values were checked against-    CITE{formula-second-difference}; and every nonsingular entry with $n\leq6$ was-    compared with NumPy's independent singular-value decomposition.+    a dry run computed all 351 entries; the Frank determinants were checked to be+    $1$; the Redheffer determinants were checked against the Mertens function; the+    Clement characteristic polynomials were checked against the eigenvalues $\pm(n-1),\pm(n-3),\ldots$;+    the second-difference values were checked against CITE{formula-second-difference};+    and every nonsingular entry with $n\leq6$ was compared with NumPy's independent+    singular-value decomposition. Display properties:   number-header: $\kappa_2(A_n)$ Numbers:-- params:-    family: hilbert-    n: '2'-  number: '19.28147006790397144831792337992132252333679086358732323389454148327244519544802785387898421871597525'-- params:-    family: hilbert-    n: '3'-  number: '524.0567775860608178707828459277826389282027210537370700223818857983732045183786337245287538826047870'-- params:-    family: hilbert-    n: '4'-  number: '15513.73873893258822257658341228279054988393607737835414587443915566453026377654442465109069584358851'-- params:-    family: hilbert-    n: '5'-  number: '476607.2502425608093811447795196152046928663241812939419678400825203841930782325253793674456254152353'-- params:-    family: hilbert-    n: '6'-  number: '14951058.64013121677509821491654659705160980645238264929146441574650661616442759402790751975685102803'-- params:-    family: hilbert-    n: '7'-  number: '475367354.9881789925590488183566092181350674661594968973815837982788531612385102004842230501459776533'-- 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