$\circ$ MURAKAMI HIROSHI TOKYO METROPOLITAN COLLEGE, MANAGEMENT AND INFORMATION monic $P$ (x $f(x=0$ (pre-conditioning $f(x/p$ (

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1 Title 一変数代数方程式の行列固有値解法について Author(s 村上, 弘 Citation 数理解析研究所講究録 (2004, Issue Date URL http//hdlhandlenet/2433/25949 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 $\circ$ MURAKAMI HIROSHI TOKYO METROPOLITAN COLLEGE, MANAGEMENT AND INFORMATION monic $P$ (x $f(x=0$ (pre-conditioning $f(x/p$ (x $f(x=0$ $P$ (x $f$ (x $f$(x $P(x$ Smith $f(x/p(x$ Durand-Kerner $P$ (x Newton-Raphson 1 $M$ $\alpha_{1},$ $\cdots,$ $\alpha_{m}$ $m_{1},$,mm $\cdots$ $\sum_{p=1}^{m}m$ p $f(x$ $n$ $P(x= \prod_{p=1}^{m}(x-\alpha_{p}^{\prime \mathrm{n}_{p}}$ $f(x/p(x=1+ \sum_{p=1}^{m}\sum_{\ell=1}^{m_{\mathrm{p}}}\mathcal{t}_{\alpha_{\mathrm{p}},l/(x-\alpha_{p}^{\ell}}$ $\tau_{\alpha_{\mathrm{p}},\ell}$ $f(x/(p(x/(x-\alpha_{p}^{m_{p}}$ $x=\alpha_{p}$ Taylor $(m_{p}-1$ $\tau_{\alpha_{\mathrm{p}},m_{p}-k}$ $f(x$ $P(x/(x-\alpha_{p}^{m}$ p $x=\alpha_{p}$ $(m_{p}-1$ Taylor $(m_{p}-1$, $m_{p}-1$ $\psi_{p}^{(k}(\alpha_{p}/k!,$ =0, $\sum_{q(\neq p\alpha_{q}} \frac{m}{x-}arrow$ $(1/k!\psi_{p}^{(k}(\alpha_{p}$ $[k]$ $\psi_{p}$ (x $\psi_{p}(x\equiv f(x/(p(x/(x-\alpha_{p}^{m_{p}}$ $B_{p}(x=- \mathrm{r}_{\alpha_{p}^{\mathrm{n}_{\mathrm{p}}}}x =\prod_{q(\neq p} Px(x-\alpha_{q}^{m_{q}}$ $\tau_{\alpha_{\mathrm{p}},m_{p}-k}=$ $G_{p}(x \equiv\frac{d}{dx}1$ Og $B_{p}(x=$ $k=0,$ $\cdots,$ $(m_{p}-1$ $\psi_{p}(x\equiv f(x/b_{p}$ (x $\tau_{\alpha_{p},m_{p}-k}=$ $\{\tau_{\alpha_{p},\ell}\}$ $\psi_{p}^{\langle k}(x\equiv\phi_{p}^{[k]}(x/b_{p}(x$ $\phi_{p}^{[k]}(x$ $\phi$ ( $\phi_{p}^{[0]}(x=f$ $\phi_{p}^{[k+1]}(x=$ (x $( \frac{d}{dx}\phi_{p}^{[k]}(x-\phi_{p}^{[k]}$ (x $G$ $\phi_{p}^{[k]}(x$ (x $f^{(k}(x$ $f(x$ - $G_{p}^{(k}(x$ $G_{p}(x$ - $G_{p}^{(k}(x=(-1^{k}k! \sum_{q(\neq p(x-}^{l}\ovalbox{\tt\small REJECT}_{\alpha_{q}}^{m}$ $\{\tau_{\alpha_{p},\ell}\}$ $\tau_{\alpha_{p},m_{p}-k}=(1/k!\psi_{p}^{(k}(\alpha_{p}$ $\psi_{p}^{(k}(x=\phi_{p}^{[k]}(x/b_{p}$ $\alpha_{p}$ $\phi_{p}^{[k]}$ (x (x 5 ( murakamiotmcaacjp XXXI-I

3 $\phi_{p}^{[3}$ 1 $\phi_{p}^{[4}$ $\phi_{\mathrm{p}}^{[0}](x=f(x$ $\phi_{p}^{[2}$1 $(x=f^{(2}(x-2g_{p}(xf^{(1}(x+(g_{p}^{2}(x-g_{p}^{(1}(xf(xr$ $\phi$pl $(x=f^{(1}(x-g_{p}(xf(x$, $(x=f^{(3}(x-3g_{p}(xf^{(2}(x+3(g_{p}^{2}(x-g_{p}^{(1}(xf^{(1}(x+(-g_{p}^{3}+3g_{p}^{(1}(xg_{p}(x-g_{p}^{(2}(xf(x$ $(x=f^{(4}(x-4g(xf^{(3}(x+(6g_{p}^{2}(x-6g_{\mathrm{p}}^{(1}(xf^{(2}(x+$ ( $-$4G (x+l2g $(xg_{p}(x-$ $4G_{p}^{(2}(xf^{(1}(x+(G_{p}^{4}(x-6G\mathrm{p}(1 (xg_{\mathrm{p}}^{2}+4g_{p}^{(2}(x+3(g_{\mathrm{p}}^{(1}(x^{2}-g_{p}^{(3}(xf(x$ 2 -ffi Lagrange $\Re\backslash$ monic $n$ $f(x=0_{\backslash }$ Lagrange $M$ $\alpha_{1},$ $\cdots,$ $\alpha_{m}$ $m_{1},$ $\cdots,$ $m_{m}$ $\sum_{p=1}^{m}m$ p $n$ $P(x=$ $\prod_{p=1}^{m}(x-\alpha_{p}^{m_{p}}$ $f(x/p(x=1+ \sum_{p=1}^{m}\sum_{l=1}^{m_{p}}\tau_{\alpha_{p}},\ell/(x-\alpha_{p}$ $f$ $M$ $n$ ( $\varphi_{\alpha_{p},\mathit{1}}(x\equiv 1/(x-\alpha_{p}^{t},$ $1\leq\ell\leq m_{p}$ $\mathrm{m}\mathrm{o}\mathrm{d} f(x$ 1 ( $P$ $\mathrm{m}\mathrm{o}\mathrm{d} f$ (x (x $1=- \sum_{q=1}^{m}\sum_{j=1}^{m_{q}}\tau$\mbox{\boldmath $\alpha$}q j $\varphi_{\alpha_{q},j}$ $\ell\geq 2$ $x\cdot\varphi_{\alpha_{\mathrm{p}}},\ell=\ulcorner_{x-\alpha_{\mathrm{p}}}^{\neg}=xr\sim Y+G^{=}d_{\mathrm{p}}$ $T\text{ }=\alpha_{p}\cdot\varphi_{\alpha_{p\prime}\ell}+1\cdot\varphi_{\alpha_{p},\ell-1}$ $\ell=1$ $x \cdot\varphi_{\alpha_{p\prime}1}=\frac{x}{x-\alpha_{\mathrm{p}}}=\frac{\alpha}{x-}\mathrm{z}_{-+1=\alpha_{p}\cdot\varphi_{\alpha_{p},1}-\sum_{q=1}^{m}\sum_{\mathrm{j}=1}^{m_{q}}\tau_{\alpha_{q},j}\cdot\varphi_{\alpha_{q},j}}\alpha_{\mathrm{p}}$ $\varphi_{\alpha_{p},\ell}$ $P$ (x Lagrange $B_{\alpha_{p},\ell}(x$ $P$ (x 21 $n$ $P(x=x^{n}$ 1 Hessenberg $f$ (x $f(x=0$ $\{1, x,x^{2}, \cdots,x^{n-1}\}$ $x=0$ $x=\infty$ (Companion matrix balancing QR- $[1]_{\text{ }}$ $\sim$ ( 22 $n$ $\alpha_{1},$ $(x-\alpha j$ $\cdots,$ $\alpha_{n}\in \mathrm{c}$ $P(x= \prod_{p}(x-\alpha_{p}$ $\tau_{j}\equiv f(\alpha_{j}/p (\alpha j_{\text{ }}x$ $f(x/p$ (x $t_{k,j}$ $=\alpha j\delta k,j-\tau j$ $f(x=0$ Durand-Kerner 1 $f(x=0$ $n$ Lagrange $P(x/(x-\alpha j$ $\alpha_{k}$ XXXI-2

4 $-\tau_{\alpha}-\tau_{0}-\tau_{\alpha}\alpha_{\mathrm{o}}1$,,, $\gamma$ $222$ $\beta-\tau_{\beta}-\tau_{\beta,1}-\tau_{\beta,1}001$ $\gamma-\tau_{\gamma}-\tau_{\gamma,1}-\tau_{\gamma1}000$ 200 Lagrange ( $[2][3]_{\text{ }}$ $QR$ $n$- QR- $O$ $O$ Schur (\sim 1 (n2 Durand-Kerner ( Durand-Kerner ( (Durand-Kerner $n$ Smith Smith Durand-Kerner $f(x=0$ ( GCD Lagrange ( [12] 23 Lagrange $\alpha_{p}$ $m_{p}$ $\text{ }$ $\circ$ $P(x= \prod_{p=1}^{m}$ ( -\mbox{\boldmath $\alpha$}pmp\downarrow $f(x/p(x=$ $1+ \sum_{p=1}^{m}\sum_{k=1}^{m_{p}}\tau$ $p k/(x-\alpha_{p}^{k}$ \mbox{\boldmath $\alpha$} $(p, k$ $(x-\alpha_{p}^{k}$ $m_{p}$ $k=m_{p},$ $\cdots,$ $2$ $(m_{p}-1$ \mbox{\boldmath $\alpha$}p 1 0 $k=1$ $(q,j$ ( $x-\alpha_{q}$ V $\tau_{\alpha_{q},j}$ $\alpha_{p}$ {3, 2, 1} $\beta,$ $f(x$ $\alpha,$ monic 6-3, 2, 1 $P(x=$ $222$, $1$,1 00$,, $[-\tau_{\alpha}-\tau_{\alpha}-\tau_{\alpha}\alpha $333$ $\alpha_{0,-\tau_{\alpha,1}}-\tau_{\alpha,1}-\tau_{\alpha,1}01$ $-\tau\rho-\tau_{\beta}-\tau_{\beta}\beta 00$, $]$ $(x- \alpha^{3}\text{ _{ }}(\begin{array}{l}x-\beta\mathrm{g}\end{array}2\text{ }\ovalbox{\tt\small REJECT}\mathrm{A}\mathrm{a}\text{ _{ }}f(x/p(x\text{ }\mathrm{g}\beta \text{ }\mathrm{f}\mathrm{f}\text{ }4\neq \mathrm{f}\mathrm{f}\mathrm{l}\text{ }ffi_{\backslash }\text{ }\{\tau_{\alpha,3},\tau_{\alpha,2},\tau_{\alpha,1}, \tau_{\beta},,\tau\tau\}\theta_{\grave{l}}r-\bigwedge_{\coprod}$ 24 ( Hessenberg double-qr balancing $ _{\vee}$ XXXI-3

5 $\tau_{p}$ $\mathrm{z}$ $\alpha_{p}$ 201 monic $n$ $f(x=0$ $n=s+2t$ $\mathrm{j}$ $\iota$ $s$ $\mathrm{f}$ 2 $\mathrm{a}$ F 1, $\ovalbox{\tt\small REJECT}$ $i$ F $\#$ $\mathrm{a}$ $\ni\lambda$ F $\mathrm{a}$ 3 balancing Francis QR- [14] PJEberlein Norm Reducing Jacobi [14] Jacobi $\mathrm{q}\mathrm{r}$ $O$ (n2 $O$ (n3 b mcing $\{\alpha\}$ $n$ $\{\tau\}$ $D$ $Tarrow\tilde{T}=DTD^{-1}$ balancing $\tau_{j}\neq 0$ $\mathcal{t}j(d_{k}/dj=(\tau_{k}\tau_{j}^{1/2}$ $d_{j}=(\tau_{j}^{1/2}$ $\tau_{j}=0$ $t_{k,j}=\alpha_{k}\delta_{k,jj}-\tauarrow\tilde{t}k,j=\alpha k\delta k,j-\tau j(d_{k}/dj$ $\epsilon$ $4=(\epsilon^{1/2}$ $\tilde{t}$ $\{\alpha\}$ ( $f$ (x $\mathcal{t}j$ ( $f$ (x $\tilde{t}$ $\tilde{t}$ $n \tau _{2}^{2}$ 2 2 I 21\geq 0( 1 $\tilde{t}$ 1 $ \tau _{1}$ 1/2\geq O( $d_{j}= \tau j ^{1/2}$ $\tilde{t}$ balancing 4 Smith 41 $f(x/p(x=1+ \sum_{p=1}^{n}\tau_{p}/(x-\alpha_{p}\text{ }$ $1 \leq\sum_{p} \tau_{p}/(x-\alpha_{p} $ 0 $n^{*}(\leq n$ $n^{*}\neq 0$ ( $1= \sum_{p}\tau_{p}/(x-\alpha_{p} $ $n^{*}=0$ $f(x/p(x\equiv 1$ $ \tau_{p}/(x-\alpha_{p} \geq 1/n^{*}$ $ x-\alpha_{p} \leq n^{*} \tau_{p} $ $q$ $ x-\alpha_{q} $ $x=\alpha_{p}$ $\tau_{\mathrm{p}}=0$ \mbox{\boldmath $\tau$}p=0 $p$ $1 \leq\sum_{q} \tau_{q}/(x-\alpha_{q} \leq(1/\min_{q} x-\alpha_{q} \sum_{q} \tau_{q} $ $ x- \alpha_{q} \leq\sum_{q} \tau_{q} = \tau _{1}$ $\alpha_{p}$ XXXI-4

6 , $\backslash \cdot$ $\alpha_{\mathrm{p}}$ 202 $ x-\alpha_{p} \leq n^{*} \tau_{p} $ $p$ $ x-\alpha_{q} \leq \tau _{1}$ $q$ $f(x=0$ $p$ $\leq n" \tau_{p} $ lx-\mbox{\boldmath $\alpha$}pl $\leq \tau _{1}$ $q$ lx-\mbox{\boldmath $\alpha$}ql $\{\tau\}$ $t$ $F$ (x, $t$ $/P(x$ $F$ (x, $\mathrm{o}$ $=P$ (x, $F$ $1$ (x, $=f$ (x ( $F(x,$ $t\equiv P(x+t(f(x-P(x_{\text{ }}$ $t$ 0 1 Gershgorin ( $ x-\alpha_{q} \leq \tau _{1}$ Smith [12] Gershgorin $ \tau_{p} $ $n^{*}$ $ \tau_{p} $ $n$ Smith Smith 42 $f(x/p(x=1+ \sum_{p=1}^{m}\sum_{k=1}^{m_{p}}\tau$ $p k/(x-\alpha_{p}^{k}$ \mbox{\boldmath $\alpha$} $1 \leq\sum_{p=1}^{m}(\sum_{k=1}^{m_{p}} \tau_{\alpha_{p},k}/(x-\alpha_{p}^{k} \circ$ M $\{\alpha_{p},p$ =1, $M\}$ $\tau_{\alpha_{p},k}$, $k=1,$ $\cdots,$ $m$ p 0 $M^{*}(\leq M$ M*=0 $f(x/p(x\equiv 1$ $M^{*}\neq 0$ $\sum_{k=1}^{m_{p}} \tau_{\alpha_{p},k}/(x-\alpha_{p}^{k} \geq 1/M^{*}$ $\tau_{\alpha_{p},k}$, $k=1,$ $\cdots,m$p 0 $p$ ( $r\equiv x-\alpha_{p} >0,a_{k}\equiv M^{*} \tau_{\alpha_{p},k} \geq 0$ $g(r \equiv 1-\sum_{k=1}^{m_{p}}a_{k}/r^{k}\leq 0$ $a_{k}$ 0 $g$(r 0 $g(r<0$ $g(r=0$ $g(r\leq 0$ r\leq Rp $ x-\alpha_{p} \leq b$ $R_{p}$ Smith $R_{\mathrm{p}}$ $g(r=0$ $k=1,$ $\cdots,$ $m$p} 0 $m_{p}^{*}$ 0 $1= \sum_{k=1}^{m_{p}}a_{k}/r_{p}^{k}$ $a_{k}/r_{p}^{k}\geq 1/m_{p}^{*}$ $R_{p}\leq(m_{\mathrm{p}}^{*}a_{k}^{1/k}$ $R_{p}$ $\max_{k}\{(m_{p}^{*}a_{k}^{1/k}\}$ $g(r$ Newton-Raphson $R_{p}$ $M^{*},$ $m_{p}^{*}$ $M,$ $m_{p}$ Smith $g$(r ($r>0$ $a_{k}$ $a_{k}$ $a_{k}$ Smith $\alpha_{p}$ $f$(x $m_{p}$ $f(x/p$(x $x=\alpha_{p}$ $k=$ $1,$ $\cdots,$ $m_{p}\}$ 0 Smith 0 $\tau_{p,k},$ { = $\cdots$ 1,, $m_{p}$ } 0 0 $\max_{k}\{(m_{p}^{*}a_{k}^{1/k}\}$ ( Smith $a_{k}arrow 0,$ =1,, $m_{p}$ 0 { $a_{k},$ { $\tau_{\alpha_{\mathrm{p}},k}$, XXXI-5

7 $\tau_{\alpha_{p}}$,k k} 203 $\alpha_{p}$ $f$ (x $L(0<L\leq m_{p}$ $\{\tau_{\alpha_{p},\mathrm{b}}k$ =1,, $f(x/p$ (x $x=\alpha_{p}$ $m_{p}-l$ $m_{p}\}$ $k=m_{p}-$ L 0 $L$ $\mathrm{t}$ $k=m_{p}-l+1,$ $\cdots,$ $m$ p 0 $L(1<L<m_{p}$ 0 $\alpha_{p}$ $f$ (x T Smith $L$ $R_{p}$ $ x-\alpha_{p} \leq R_{\mathrm{p}}$ $x=\alpha_{p}$ $\alpha_{p}$ Smith $R_{p}$ $f(x=0$ ( ( 0 $f(x/p$(x $\tau_{\alpha_{p}}$ { $t^{k}\tau_{\alpha_{p}}$ {,k} $F(x, t/p$ (x $F$ (x, $\mathrm{o}$ $=P$ (x, $F$ (x, $1$ $t$ $=f$ (x $t=0$ $t=1$ $p$ Smith $tr_{p}$ Gershgorin ( Smith [12] (Smith $R_{p}$ $\{\tau_{\alpha_{\mathrm{p}},k},$ =1,, $m_{p}\}$ $h_{p,k}(t$ (t $h_{p,k}$ $t$ $[0, 1]$ $[0, 1]$ $h_{p,k}(t=t$ 5 $f(x/p(x=1+ \sum_{j=1}^{n}\tau_{j}/(x-\alpha_{j}$ $\tau_{j}=f(\alpha_{j}/p (\alpha_{j}_{\text{ }}$ $\alpha_{k}$ l+\sigma jn $=1\tau j/(x-\alpha j=0$ $\tau_{k}=0$ $f(x=0$ $P(x=0$ $\{\alpha\}$ $\{\tau\}$ O( 0 ($f(x=0$ $P(x$ pre-conditioning Newton-Raphson Newton-Raphson $g(x \equiv 1+\sum_{j}\tau_{j}/(x-\alpha_{j}$ Raphson $g$ Newton- $x \vdash x-g(x/g (x=x+(1+\sum_{j}\tau_{j}/(x-\alpha_{j}/(\sum_{j}\tau_{j}/(x-\alpha_{j}^{2}$ $\tau_{j}\neq 0$ ( $\alpha_{j}$ 0 XXXI-6

8 $\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{o}\mathrm{n},\mathrm{v}\mathrm{o}\mathrm{l}11,\mathrm{p}\mathrm{p}1-20,1994$ $\alpha_{k}$ 204 $\{\tau\}$ $(x-\alpha_{k}$ $x^{(s+0}=\alpha k$ $-\tau_{k}-$ (x(s - $x=\alpha_{k}-$ \mbox{\boldmath $\tau$}k $-(x- \alpha_{k}\sum_{\mathrm{j}(\neq k} \tau_{j}/(x-\alpha_{j}$ $(\neq k\tau_{j}/(x^{(s}-\alpha_{j}$ $\alpha_{k}\sum$ j $\tau$ 1 $x^{(1}=\alpha_{k}-\tau k$ Durand-Kerner $\tau$ 2 $\theta^{(2}\equiv 1-\Sigma j(\neq k\tau j/(x^{(1}-\alpha_{j}$ $x^{(2}=\alpha_{k}-\tau k\theta^{(2}$ 3 $\theta^{(3}\equiv 1-\theta^{(2}\sum_{j(\neq k} \tau_{j}/(x^{(2}-\alpha_{j},$ $\theta^{(1}\equiv 1$ 1 $x^{(3}=\alpha_{k}-\tau_{k}\theta$ (3 $x^{(1}=\alpha_{k}-\tau_{k}\theta^{(1}$ $\theta^{(s+1}\equiv 1-\theta^{(s}\sum_{j(\neq k} \tau_{j}/(x^{(\epsilon}-\alpha_{j}\acute{}x^{(s+1}=\alpha_{k}-\tau_{k}\theta^{(s+1}$ $(s+1$ $s$ $\theta^{(\epsilon},x^{(\epsilon}$ [1] Alan Edelman and H Murakami, Polynomial Roots from Companion Matrix Eigenvalues, Math C0mp,v64,n0210,p ,Apri1,(1995 [2] Steven Fortune, Polynomial root finding using iterated eigenvalue computation, Proceedings of $\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{s}\mathrm{a}\mathrm{m},\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{c}$ the ACM 2001,ppl2l-l28 ( [3] Steven Fortune, Convergence analysis of an iterated eigenvalue polynomial root finding algorithm, URL citeseernjnec $=$ $\mathrm{c}\mathrm{o}\mathrm{m}/469859$ html [4] Gene HGolub and Charles FVan Loan, Matrix Computations, 3rd ed, The Johns Hopkins University, Press, 1996 ( [5] Peter Henrici, Applied and Computational Complex Analysis, VoII John Wiley and Sons $6$ -Polynomials $\mathrm{i}\mathrm{n}\mathrm{c},1974$ $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{p}$ ( [6] Dinesh Manocha, Algorithms for computing selected solutions of polynomial equations, JSymbolic ( [7] Dinesh Manocha and Shankar Krishnan, Solving Algebraic Systems using Matrix Computations $\mathrm{b}\mathrm{u}\mathrm{u}\mathrm{e}\mathrm{t}\mathrm{i}\mathrm{n},\mathrm{v}\mathrm{o}\mathrm{l}30,\mathrm{n}\mathrm{o}4,\mathrm{p}\mathrm{p}4-21,1996$ SIGSAM [8] Arnold Neumaier, A Gerschgorin-type theorem for zeros of polynomials [9] VPan, Solving a polynomial equation some history and recent progress, SIAM $\mathrm{r}\mathrm{e}\mathrm{v}\mathrm{i}\mathrm{e}\mathrm{w},\mathrm{v}\mathrm{o}\mathrm{l}39,\mathrm{n}\mathrm{o}2,\mathrm{p}\mathrm{p} ,1997$ ( [10] Miodrag SPetkovic, Caxsten Carstensen and Miroslav Trajkovic, Weierstrass formula and zerofinding metbods(1995, URL $=$ $\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{j}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{m}/\mathrm{p}\mathrm{e}\mathrm{t}\mathrm{k}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{c}95\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{h}\mathrm{t}\mathrm{m}\mathrm{l}$ [11] WHPress, BPFlannery, SATaukolsky and WTVetterling, Numerical Recipes in Pascal The - Art of Scientific Computing, Cambridge University Press, 1989 [12] BTSmith, Error bounds for the zeroes of a polynomial based upon Gershgorin s theorem, JACM, ,1970 $\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s},\mathrm{o}\mathrm{x}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{d},1965$ [13] J Wilkinson, The Algebraic Eigenvalue Problem, Clarendom ( [14] JHWilkinson and CReinsch, Handbook for Automatic Computation VolII, Linear Algebra, $\mathrm{q}\mathrm{r}$ $\mathrm{q}\mathrm{r}$ Springer-Verlag,1971 ( Francis double Norm Reducing Jacobi Alg01-60 XXXI-7

Archimedean Spiral 1, ( ) Archimedean Spiral Archimedean Spiral ( $\mathrm{b}.\mathrm{c}$ ) 1 P $P$ 1) Spiral S

$Archimedean Spiral 1, ( ) Archimedean Spiral Archimedean Spiral ( $\mathrm{b}.\mathrm{c}$ ) 1 P $P$ 1) Spiral S$ Title 初期和算にみる Archimedean Spiral について ( 数学究 ) Author(s) 小林, 龍彦 Citation 数理解析研究所講究録 (2000), 1130: 220-228 Issue Date 2000-02 URL http://hdl.handle.net/2433/63667 Right Type Departmental Bulletin Paper Textversion