Abstract We give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between T † ⁢ T ⁢ ( X n ) {T^{\dagger}T(X_{n})} and T * ⁢ T ⁢ ( X n ) {T^{*}T(X_{n})} , and the principal angles between X n ∩ ( 𝒩 ⁢ ( T ) ∩ X n ) ⊥ {X_{n}\cap(\mathcal{N}(T)\cap X_{n})^{\perp}} and ( 𝒩 ⁢ ( T ) + X n ) ∩ 𝒩 ⁢ ( T ) ⊥ {(\mathcal{N}(T)+X_{n})\cap\mathcal{N}(T)^{\perp}} . At the end, we consider several specific cases and examples to further illustrate our theorems.

中文翻译：

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不适定问题的最小二乘投影方案的因式分解

摘要 我们给出了最小二乘投影方案的分解公式，从中可以推导出新的收敛条件以及估计收敛速度的公式。我们证明了该方法的收敛性（包括收敛速度）可以完全由 T † ⁢ T ⁢ ( X n ) {T^{\dagger}T(X_{n})} 和 T 之间的主夹角决定* ⁢ T ⁢ ( X n ) {T^{*}T(X_{n})} ，以及 X n ∩ ( 𝒩 ⁢ ( T ) ∩ X n ) ⊥ {X_{n}\cap( \mathcal{N}(T)\cap X_{n})^{\perp}} 和 ( 𝒩 ⁢ ( T ) + X n ) ∩ 𝒩 ⁢ ( T ) ⊥ {(\mathcal{N}(T)+ X_{n})\cap\mathcal{N}(T)^{\perp}} 。最后，我们考虑几个具体的案例和例子来进一步说明我们的定理。