Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems

Abstract

Given a polynomial system f that is associated with an isolated singular zero ξ whose Jacobian matrix is of corank one, and an approximate zero x that is close to ξ, we propose an improved two-step Newton’s method for refining x to converge to ξ with quadratic convergence. Our new approach is based on a closed-form basis of the local dual space and a recursive reduction of the simple multiple zero. By avoiding solving several least-squares problems which appeared in the previous methods, an overall 2 ×-5 × acceleration is achieved. The proof of the quadratic convergence of proposed iterations is also simplified significantly. Numerical experiments demonstrate up to 100 × speed-up when we replace the least-squares-solving calculations with closed-form solutions for refining approximate singular solutions of large-size problems (1000 equations and 1000 variables).

Appendix

We prove below conditions (16), (17), (18), and (46).

Proof Proof of ( 16 )

. It is easy to check that

$$ \mathbf{d}^{\alpha}_{\xi}\left( (X-\xi)^{\beta}\right) =\left\{\begin{array}{rl} 1, & \alpha=\beta, \\ 0, & \text{otherwise.} \end{array} \right. $$

Since the order of every differential monomial in Δ_k is bounded above by k, we derive

$$ {{\varDelta}}_{k}\left( (X-\xi)^{\beta}\right) =0, ~\text{if}~|\beta|>k. $$

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Proof Proof of ( 17 )

. Since (X − ξ)^βf_i(X) ∈ I_f for i = 1,…,n, we have

$$ \begin{array}{@{}rcl@{}} 0&=&{{\varLambda}}_{k}\left( (X-\xi)^{\beta}f(X)\right)\\ &=&{{\varDelta}}_{k}\left( (X - \xi)^{\beta}f(X)\right) +a_{k,1}d_{1}\left( (X - \xi)^{\beta}f(X)\right) +\cdots+a_{k,n}d_{n}\left( (X - \xi)^{\beta}f(X)\right). \end{array} $$

Since f(ξ) = 0, we have $d_j\left ((X-\xi )^{\beta }f(X)\right )=0$ for j = 1,…,n. Therefore, we derive

$$ {{\varDelta}}_{k}\left( (X-\xi)^{\beta}f(X)\right) =0, ~~\text{if}~|\beta|>0. $$

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Proof Proof of ( 18 )

. Let ${{\varPhi }}[\mathbf {a}_k]={\sum }_{i=1}^n a_{k,i}{{\varPsi }}_i$, where ${{\varPsi }}_{i}:\mathfrak {D}_{\xi }\rightarrow \mathfrak {D}_{\xi }$ be the morphism that satisfies ${{\varPsi }}_{i}(d_1^{\alpha _1}{\cdots } d_n^{\alpha _n})=d_{i}^{\alpha _{i}+1} {\cdots } d_n^{\alpha _n}$ if α₁ = ⋯ = α_i− 1 = 0 and 0 otherwise for i = 1,…,n, then (5) and (6) can be rewritten into

$$ \begin{array}{@{}rcl@{}} {{\varDelta}}_{k}&=&\sum\limits_{j=1}^{k-1} {{\varPhi}}[\mathbf{a}_{j}]({{\varLambda}}_{k-j}), \\ {{\varLambda}}_{k}&=&{{\varDelta}}_{k}+{{\varPhi}}[\mathbf{a}_{k}](1). \end{array} $$

Consequently, Δ_k and Λ_k can be written into a sum of homogenous terms formulated by ${\prod }_{j=1}^k {{\varPhi }}[\mathbf {a}_j]^{n_j}(1)$, where n_j ≥ 0 and ${\sum }_{j=1}^k n_j\leq k$. We can check that

$$ \prod\limits_{j=1}^{k} {{\varPhi}}[\mathbf{a}_{j}]^{n_{j}}(1)\left( [v^{*}(X-\xi)]^{l}\right) =\left\{\begin{array}{rl} {\prod}_{j=1}^{k}(v^{*}\mathbf{a}_{j})^{n_{j}}, & {\sum}_{j=1}^{k} n_{j}=l, \\ 0, & \text{otherwise.} \end{array} \right.$$

Since Φ[a₁]^k is the only pure term of Φ[a₁] in Δ_k (other terms are mixed with at least one Φ[a_j] for j = 2,…,k − 1), and ${v_n}^{*}\mathbf {a}_1=1$, ${v_n}^{*}\mathbf {a}_j=0$ according to (7), we derive

$$ {{\varDelta}}_{k}\left( \left[{v_{n}}^{*}(X-\xi)\right]^{l}\right) =\left\{\begin{array}{rl} 1, & \text{if}~k=l, \\ 0, & \text{otherwise}. \end{array} \right. $$

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Proof Proof of ( 46 )

. Since $\|{v^{\prime }_n}^{\ast }(x^{\prime }-\xi )\|\leq \|{v^{\prime }_n}^{\ast }\|\|x^{\prime }-\xi \|=O(\epsilon )$, we get

$$ \begin{array}{@{}rcl@{}} \left[{v^{\prime}_{n}}^{\ast}(X-x^{\prime})\right]^{k}\!& = &\!\left[{v^{\prime}_{n}}^{\ast}(X-\xi)+{v^{\prime}_{n}}^{\ast}(\xi-x^{\prime})\right]^{k}\\ \!& = &\!\left[{v^{\prime}_{n}}^{\ast}(X - \xi)\right]^{k} + k\left[{v^{\prime}_{n}}^{\ast}(X - \xi)\right]^{k-1} \left[{v^{\prime}_{n}}^{\ast}(\xi - x^{\prime})\right] + O(\epsilon^{2}) \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} \!& = &\!\left[{v^{\prime}_{n}}^{\ast}(X-\xi)\right]^{k} +O(\epsilon). \end{array} $$

(48)

Similar to the proof of (18), $\hat {{{\varDelta }}}_{\mu -1}$ and $\hat {{{\varLambda }}}_{\mu -1}$ can be written as a sum of homogenous terms formulated by ${\prod }_{j=1}^{\mu -1} {{\varPhi }}[\hat {\mathbf {a}}_j]^{n_j}(1)$, where n_j ≥ 0 and ${\sum }_{j=1}^{\mu -1} n_j\leq \mu -1$. According to (33) and (34), we know that $|{v^{\prime }_n}^{*}\hat {\mathbf {a}}_1|=1-O(\epsilon )$ and $|{v^{\prime }_n}^{*}\hat {\mathbf {a}}_j|=O(\epsilon )$, so we have

$$ {\prod}_{j=1}^{\mu-1} {{\varPhi}}[\hat{\mathbf{a}}_{j}]^{n_{j}}(1)\left( [{v^{\prime}_{n}}^{*}(X-\xi)]^{k}\right) =\left\{\begin{array}{rl} ({v^{\prime}_{n}}^{*}\hat{\mathbf{a}}_{1})^{k}, & {\sum}_{j=1}^{\mu-1} n_{j}=k \text{ and } n_{1}=k, \\ O(\epsilon), & {\sum}_{j=1}^{\mu-1} n_{j}=k \text{ and } n_{1}<k, \\ 0, & \text{otherwise.} \end{array} \right. $$

(49)

Since ${{\varPhi }}[\hat {\mathbf {a}}_1]^{\mu -1}$ is the only pure term of ${{\varPhi }}[\hat {\mathbf {a}}_1]$ in $\hat {{{\varLambda }}}_{\mu -1}$ (other terms are mixed with at least one ${{\varPhi }}[\hat {\mathbf {a}}_j]$ for j = 2,…,μ − 1), combining (48) and (49), we derive

$$ \hat{{{\varLambda}}}_{\mu-1}\left( \left[{v^{\prime}_{n}}^{\ast}(X-x^{\prime})\right]^{k}\right)= \left\{\begin{array}{rl} O(\epsilon), & ~~~ k\leq\mu-2, \\ ({v^{\prime}_{n}}^{\ast}\hat{\mathbf{a}}_{1})^{\mu-1} +O(\epsilon), & ~~~ k=\mu-1. \end{array} \right. $$

For k = μ, combining (47) and (49), we derive

$$ \begin{array}{@{}rcl@{}} &&\hat{{{\varLambda}}}_{\mu-1}\left( \left[{v^{\prime}_{n}}^{\ast}(X-x^{\prime})\right]^{\mu}\right)\\ &=&\hat{{{\varLambda}}}_{\mu-1}\left( \left[{v^{\prime}_{n}}^{\ast}(X-\xi)\right]^{\mu}\right) +\hat{{{\varLambda}}}_{\mu-1}\left( {\mu}\left[{v^{\prime}_{n}}^{\ast}(X-\xi)\right]^{\mu-1} \left[{v^{\prime}_{n}}^{\ast}(\xi-x^{\prime})\right]\right)+O(\epsilon^{2})\\ &=& 0+\mu{v^{\prime}_{n}}^{\ast}(\xi-x^{\prime})({v^{\prime}_{n}}^{\ast}\hat{\mathbf{a}}_{1})^{\mu-1} +O(\epsilon^{2}). \end{array} $$

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