nonlinear-solvers

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Nonlinear Solvers

非线性求解器

Goal

目标

Provide a universal workflow to select a nonlinear solver, configure globalization strategies, and diagnose convergence for root-finding, optimization, and least-squares problems.

提供通用工作流，为根求解、优化和最小二乘问题选择非线性求解器、配置全局策略并诊断收敛情况。

Requirements

要求

Python 3.10+
NumPy (for Jacobian diagnostics)
SciPy (optional, for advanced analysis)

Python 3.10+
NumPy（用于Jacobian矩阵诊断）
SciPy（可选，用于高级分析）

Inputs to Gather

需要收集的输入信息

Input	Description	Example
Problem type	Root-finding, optimization, least-squares	`root-finding`
Problem size	Number of unknowns	`n = 10000`
Jacobian availability	Analytic, finite-diff, unavailable	`analytic`
Jacobian cost	Cheap or expensive to compute	`expensive`
Constraints	None, bounds, equality, inequality	`none`
Smoothness	Is objective/residual smooth?	`yes`
Residual history	Sequence of residual norms	`1,0.1,0.01,...`

输入项	描述	示例
问题类型	根求解、优化、最小二乘	`root-finding`
问题规模	未知量数量	`n = 10000`
Jacobian矩阵可用性	解析形式、有限差分、不可用	`analytic`
Jacobian矩阵计算成本	低或高	`expensive`
约束条件	无、边界约束、等式约束、不等式约束	`none`
平滑性	目标函数/残差是否平滑？	`yes`
残差历史	残差范数序列	`1,0.1,0.01,...`

Decision Guidance

决策指南

Solver Selection Flowchart

求解器选择流程图

Is Jacobian available and cheap?
├── YES → Problem size?
│   ├── Small (n < 1000) → Newton (full)
│   └── Large (n ≥ 1000) → Newton-Krylov
└── NO → Is objective smooth?
    ├── YES → Memory limited?
    │   ├── YES → L-BFGS or Broyden
    │   └── NO → BFGS
    └── NO → Anderson acceleration or Picard

Is Jacobian available and cheap?
├── YES → Problem size?
│   ├── Small (n < 1000) → Newton (full)
│   └── Large (n ≥ 1000) → Newton-Krylov
└── NO → Is objective smooth?
    ├── YES → Memory limited?
    │   ├── YES → L-BFGS or Broyden
    │   └── NO → BFGS
    └── NO → Anderson acceleration or Picard

Quick Reference

快速参考表

Problem Type	First Choice	Alternative	Globalization
Small root-finding	Newton	Broyden	Line search
Large root-finding	Newton-Krylov	Anderson	Trust region
Optimization	L-BFGS	BFGS	Wolfe line search
Least-squares	Levenberg-Marquardt	Gauss-Newton	Trust region
Bound constrained	L-BFGS-B	Trust-region reflective	Projected

问题类型	首选方案	替代方案	全局策略
小规模根求解	Newton	Broyden	线搜索
大规模根求解	Newton-Krylov	Anderson	信赖域
优化问题	L-BFGS	BFGS	Wolfe线搜索
最小二乘问题	Levenberg-Marquardt	Gauss-Newton	信赖域
带边界约束问题	L-BFGS-B	信赖域反射法	投影法

Script Outputs (JSON Fields)

脚本输出（JSON字段）

Script	Key Outputs
`scripts/solver_selector.py`	`recommended` , `alternatives` , `notes`
`scripts/convergence_analyzer.py`	`converged` , `convergence_type` , `estimated_rate` , `diagnosis`
`scripts/jacobian_diagnostics.py`	`condition_number` , `jacobian_quality` , `rank_deficient`
`scripts/globalization_advisor.py`	`strategy` , `line_search_type` , `trust_region_type` , `parameters`
`scripts/residual_monitor.py`	`patterns_detected` , `alerts` , `recommendations`
`scripts/step_quality.py`	`ratio` , `step_quality` , `accept_step` , `trust_radius_action`

脚本	核心输出
`scripts/solver_selector.py`	`recommended` , `alternatives` , `notes`
`scripts/convergence_analyzer.py`	`converged` , `convergence_type` , `estimated_rate` , `diagnosis`
`scripts/jacobian_diagnostics.py`	`condition_number` , `jacobian_quality` , `rank_deficient`
`scripts/globalization_advisor.py`	`strategy` , `line_search_type` , `trust_region_type` , `parameters`
`scripts/residual_monitor.py`	`patterns_detected` , `alerts` , `recommendations`
`scripts/step_quality.py`	`ratio` , `step_quality` , `accept_step` , `trust_radius_action`

Workflow

工作流

Characterize problem - Identify type, size, Jacobian availability
Select solver - Run
```
scripts/solver_selector.py
```
Choose globalization - Run
```
scripts/globalization_advisor.py
```
Analyze Jacobian - If available, run
```
scripts/jacobian_diagnostics.py
```
Monitor residuals - During solve, use
```
scripts/residual_monitor.py
```
Analyze convergence - Run
```
scripts/convergence_analyzer.py
```
Evaluate steps - For trust region, use
```
scripts/step_quality.py
```

问题特征分析 - 确定问题类型、规模、Jacobian矩阵可用性
选择求解器 - 运行
```
scripts/solver_selector.py
```
选择全局策略 - 运行
```
scripts/globalization_advisor.py
```
分析Jacobian矩阵 - 若矩阵可用，运行
```
scripts/jacobian_diagnostics.py
```
监控残差 - 求解过程中使用
```
scripts/residual_monitor.py
```
分析收敛情况 - 运行
```
scripts/convergence_analyzer.py
```
评估步长质量 - 针对信赖域方法，使用
```
scripts/step_quality.py
```

Conversational Workflow Example

对话式工作流示例

User: My Newton solver for a phase-field simulation is converging very slowly. After 50 iterations, the residual only dropped from 1 to 0.1.

Agent workflow:

Analyze convergence:

bash

python3 scripts/convergence_analyzer.py --residuals 1,0.8,0.6,0.5,0.4,0.3,0.2,0.15,0.12,0.1 --json

Check globalization strategy:

bash

python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json

Recommend: Switch to trust region with Levenberg-Marquardt regularization, or use Newton-Krylov with better preconditioning.

用户：我用于相场模拟的Newton求解器收敛非常慢。50次迭代后，残差仅从1降到0.1。

Agent工作流:

分析收敛情况:

bash

python3 scripts/convergence_analyzer.py --residuals 1,0.8,0.6,0.5,0.4,0.3,0.2,0.15,0.12,0.1 --json

检查全局策略:

bash

python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality ill-conditioned --previous-failures 0 --json

推荐方案：切换到带Levenberg-Marquardt正则化的信赖域方法，或使用带更好预条件子的Newton-Krylov方法。

Pre-Solve Checklist

求解前检查清单

CLI Examples

CLI示例

bash

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bash

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Select solver for large unconstrained optimization

为大规模无约束优化选择求解器

python3 scripts/solver_selector.py --size 50000 --smooth --memory-limited --json

Select solver for a small nonlinear least-squares (data-fitting) problem

为小型非线性最小二乘（数据拟合）问题选择求解器

python3 scripts/solver_selector.py --problem-type least-squares --size 6 --jacobian-available --smooth --json

Analyze convergence from residual history

根据残差历史分析收敛情况

python3 scripts/convergence_analyzer.py --residuals 1,0.1,0.01,0.001,0.0001 --tolerance 1e-6 --json

Diagnose Jacobian quality

诊断Jacobian矩阵质量

python3 scripts/jacobian_diagnostics.py --matrix jacobian.txt --json

Get globalization recommendation

获取全局策略推荐

python3 scripts/globalization_advisor.py --problem-type optimization --jacobian-quality good --json

Globalization for a distant initial guess (favors trust region)

针对远离解的初始猜测的全局策略（优先信赖域）

python3 scripts/globalization_advisor.py --problem-type root-finding --jacobian-quality good --far-from-solution --json

Monitor residual patterns

监控残差模式

python3 scripts/residual_monitor.py --residuals 1,0.8,0.9,0.7,0.75,0.6 --target-tolerance 1e-8 --json

Evaluate step quality for trust region

评估信赖域方法的步长质量

python3 scripts/step_quality.py --predicted-reduction 0.5 --actual-reduction 0.4 --step-norm 0.8 --gradient-norm 1.0 --trust-radius 1.0 --json

undefined

python3 scripts/step_quality.py --predicted-reduction 0.5 --actual-reduction 0.4 --step-norm 0.8 --gradient-norm 1.0 --trust-radius 1.0 --json

undefined

Error Handling

错误处理

Error	Cause	Resolution
`problem_size must be positive`	Invalid size	Check problem dimension
`problem_size (...) exceeds maximum (...)`	Size above 10 billion cap	Re-check the unit/value
`constraint_type must be one of...`	Unknown constraint	Use: none, bound, equality, inequality
`problem_type must be one of...`	Unknown problem type	Use: root-finding, optimization, least-squares
`residuals must be non-negative`	Invalid residual data	Check residual computation
`residuals must be finite`	NaN/Inf in residual data	Sanitize residual history
`residual list length (...) exceeds limit (...)`	More than 100,000 entries	Downsample the history
`Matrix file not found`	Invalid path	Verify Jacobian file exists
`Matrix file exceeds size limit ...`	Matrix file too large	Use a smaller / sparser matrix

错误信息	原因	解决方法
`problem_size must be positive`	无效规模值	检查问题维度
`problem_size (...) exceeds maximum (...)`	规模超过100亿上限	重新检查单位/数值
`constraint_type must be one of...`	未知约束类型	使用以下值：none, bound, equality, inequality
`problem_type must be one of...`	未知问题类型	使用以下值：root-finding, optimization, least-squares
`residuals must be non-negative`	无效残差数据	检查残差计算逻辑
`residuals must be finite`	残差数据包含NaN/Inf	清理残差历史数据
`residual list length (...) exceeds limit (...)`	条目超过10万条	对历史数据进行下采样
`Matrix file not found`	路径无效	确认Jacobian矩阵文件存在
`Matrix file exceeds size limit ...`	矩阵文件过大	使用更小/更稀疏的矩阵

Interpretation Guidance

解读指南

Convergence Type

收敛类型

Type	Meaning	Action
quadratic	Optimal Newton (order p ≈ 2)	Continue, near solution
superlinear	Ratios shrinking toward 0 (1 < p < 2); quasi-Newton working	Monitor for stagnation
linear	Constant contraction ratio (p ≈ 1); a small constant ratio is fast-linear, not superlinear	May improve with preconditioner
sublinear	Too slow (ratio → 1)	Change method or formulation
stagnated	No progress	Check Jacobian, preconditioner
diverged	Increasing residual	Add globalization, check Jacobian

类型	含义	操作建议
quadratic	最优Newton收敛（阶数p≈2）	继续执行，已接近解
superlinear	收缩比趋近于0（1 < p < 2）；拟Newton方法生效	监控是否出现停滞
linear	收缩比恒定（p≈1）；小恒定比属于快速线性收敛，而非超线性	可考虑添加预条件子
sublinear	收敛过慢（收缩比趋近于1）	更换方法或重新构建问题
stagnated	无进展	检查Jacobian矩阵、预条件子
diverged	残差增大	添加全局策略，检查Jacobian矩阵

Jacobian Quality

Jacobian矩阵质量

Quality	Condition Number	Action
good	< 10⁶	Standard Newton works
moderately-conditioned	10⁶ - 10¹⁰	Consider scaling
ill-conditioned	> 10¹⁰	Use regularization
near-singular	∞	Reformulate or use LM

质量等级	条件数	操作建议
good	< 10⁶	标准Newton方法可行
moderately-conditioned	10⁶ - 10¹⁰	考虑进行缩放处理
ill-conditioned	> 10¹⁰	使用正则化方法
near-singular	∞	重新构建问题或使用LM方法

Step Quality (Trust Region)

步长质量（信赖域）

Ratio ρ	Quality	Trust Radius
ρ < 0	very_poor	Shrink aggressively
ρ < 0.25	marginal	Shrink
0.25 ≤ ρ < 0.75	good	Maintain
ρ ≥ 0.75	excellent	Expand if at boundary

比值ρ	质量等级	信赖域半径操作
ρ < 0	very_poor	大幅缩小半径
ρ < 0.25	marginal	缩小半径
0.25 ≤ ρ < 0.75	good	保持半径
ρ ≥ 0.75	excellent	若步长触达边界则扩大半径

Verification checklist

验证检查清单

Do not trust a "solved" claim until these concrete artifacts are recorded:

Logged the full residual norm history and ran
```
convergence_analyzer.py --residuals <history>
```
; recorded
```
convergence_type
```
and
```
estimated_rate
```
, and confirmed
```
converged: true
```
against the actual solver tolerance (not the default
```
1e-10
```
).
Confirmed the residual sequence is monotone-decreasing or fed it to
```
residual_monitor.py
```
; recorded
```
patterns_detected
```
and verified it does NOT include
```
diverging
```
,
```
oscillating
```
,
```
plateau
```
, or
```
slow_convergence
```
while still above tolerance.
If a Jacobian is available, ran
```
jacobian_diagnostics.py --matrix J.txt
```
and recorded
```
condition_number
```
and
```
jacobian_quality
```
; for an analytic Jacobian, passed
```
--finite-diff-matrix
```
and confirmed
```
finite_diff_error
```
is below ~1e-2 (no "Large discrepancy" note).
Checked
```
rank_deficient
```
from
```
jacobian_diagnostics.py
```
is
```
false
```
(or documented why a rank-deficient/near-singular Jacobian is expected and that Levenberg-Marquardt regularization is in use).
For a trust-region solve, evaluated accepted steps with
```
step_quality.py
```
and recorded the reduction
```
ratio
```
; confirmed accepted steps have
```
ratio >= 0.25
```
(not
```
very_poor
```
/
```
poor
```
) and that the
```
trust_radius_action
```
matches the recorded ρ.
Recorded the solver and globalization actually used and confirmed they match
```
solver_selector.py
```
and
```
globalization_advisor.py
```
recommendations for the stated problem type, size, and Jacobian quality (e.g., large/expensive-Jacobian → Newton-Krylov; least-squares → Levenberg-Marquardt trust region).
Re-confirmed convergence after any change to tolerance, initial guess, or preconditioner — the convergence type can flip (e.g., quadratic → linear/stagnated) and must be re-classified, not assumed.

在记录以下具体成果前，不要轻信“已求解”的结论：

记录完整的残差范数历史，并运行
```
convergence_analyzer.py --residuals <history>
```
；记录
```
convergence_type
```
和
```
estimated_rate
```
，并确认
```
converged: true
```
与求解器实际容差匹配（而非默认的
```
1e-10
```
）。
确认残差序列单调递减，或已将其输入
```
residual_monitor.py
```
；记录
```
patterns_detected
```
，并验证在容差之上时不包含
```
diverging
```
、
```
oscillating
```
、
```
plateau
```
或
```
slow_convergence
```
。
若Jacobian矩阵可用，运行
```
jacobian_diagnostics.py --matrix J.txt
```
并记录
```
condition_number
```
和
```
jacobian_quality
```
；对于解析形式的Jacobian矩阵，传入
```
--finite-diff-matrix
```
并确认
```
finite_diff_error
```
低于约1e-2（无“Large discrepancy”提示）。
检查
```
jacobian_diagnostics.py
```
返回的
```
rank_deficient
```
为
```
false
```
（或记录为何Jacobian矩阵是秩亏/近奇异的，并确认已使用Levenberg-Marquardt正则化）。
对于信赖域求解，使用
```
step_quality.py
```
评估已接受的步长并记录收缩
```
ratio
```
；确认已接受步长的
```
ratio >= 0.25
```
（非
```
very_poor
```
/
```
poor
```
），且
```
trust_radius_action
```
与记录的ρ匹配。
记录实际使用的求解器和全局策略，并确认它们与
```
solver_selector.py
```
和
```
globalization_advisor.py
```
针对所述问题类型、规模和Jacobian矩阵质量给出的推荐一致（例如，大规模/高成本Jacobian矩阵→Newton-Krylov；最小二乘问题→Levenberg-Marquardt信赖域）。
在修改容差、初始猜测或预条件子后，重新确认收敛情况——收敛类型可能会变化（例如，二次→线性/停滞），必须重新分类，而非假设不变。

Common pitfalls & rationalizations

常见陷阱与误区

Tempting shortcut	Why it's wrong / what to do
"The residual ratio is a small constant (~0.1), so it's converging superlinearly."	A constant contraction ratio is linear, not superlinear — `convergence_analyzer.py` reports this as `linear` (annotated "fast linear"). Superlinear requires the ratio to tend to zero (order p > 1.2). Don't claim Newton-quality convergence from a flat ratio.
"It stopped without erroring, so the solver converged."	Run completion is not convergence. Check `converged` from `convergence_analyzer.py` / `residual_monitor.py` against the real tolerance; a `stagnated` or `plateau` result also "stops" but has not solved `f(x)=0` .
"Two iterations look like they're shrinking, so the rate is fine."	Order estimation needs at least 3 strictly decreasing positive residuals; with fewer, `convergence_analyzer.py` returns `unknown` /falls back to rate-only. Gather more iterations before quoting a convergence type.
"I coded the analytic Jacobian, so it must be right."	A wrong Jacobian still produces some step. Run `jacobian_diagnostics.py --finite-diff-matrix` and confirm `finite_diff_error` is small; a "Large discrepancy with finite-diff" note means the analytic Jacobian is buggy, which silently degrades Newton to linear convergence.
"Newton diverged, so I'll just shrink the global tolerance and call it close enough."	Divergence ( `convergence_type: diverged` , or `diverging` pattern) signals a bad step direction or far-from-solution start — add globalization. Run `globalization_advisor.py` (use `--far-from-solution` / report failures) and switch to a trust region or damped step instead of loosening the target.
"Trust-region step decreased the objective, so accept and expand the radius."	Acceptance and radius growth depend on the reduction ratio ρ, not just sign. `step_quality.py` only flags `expand` when ρ ≥ 0.75 and the step hit the boundary; a small positive ρ ( `marginal` ) means accept-but-shrink. Use the recorded `trust_radius_action` .
"The Jacobian is large and expensive, but full Newton is the gold standard, so I'll form it anyway."	For n ≥ 1000 or expensive Jacobians, `solver_selector.py` routes to matrix-free Newton-Krylov (JFNK) precisely because forming/factoring J is infeasible; use Jacobian-vector products plus a preconditioner instead.

诱人的捷径	错误原因/正确做法
“残差比是一个小常数（≈0.1），所以是超线性收敛。”	恒定收缩比属于线性收敛，而非超线性—— `convergence_analyzer.py` 会将其标记为 `linear` （注释为“fast linear”）。超线性收敛要求收缩比趋近于0（阶数p>1.2）。不要根据恒定比值声称达到Newton级收敛。
“求解器未报错就停止了，所以已经收敛。”	运行完成不等于收敛。检查 `convergence_analyzer.py` / `residual_monitor.py` 返回的 `converged` 是否符合实际容差； `stagnated` 或 `plateau` 结果也会“停止”，但并未解决 `f(x)=0` 。
“两次迭代看起来在收缩，所以收敛速率没问题。”	阶数估计至少需要3个严格递减的正残差；若残差数量不足， `convergence_analyzer.py` 会返回 `unknown` /仅返回速率。在引用收敛类型前，收集更多迭代数据。
“我编写了解析形式的Jacobian矩阵，所以肯定是正确的。”	错误的Jacobian矩阵仍会产生步长。运行 `jacobian_diagnostics.py --finite-diff-matrix` 并确认 `finite_diff_error` 较小；若出现“Large discrepancy with finite-diff”提示，说明解析Jacobian矩阵存在bug，会悄无声息地将Newton收敛降级为线性收敛。
“Newton方法发散了，我只要缩小全局容差就可以认为足够接近了。”	发散（ `convergence_type: diverged` 或 `diverging` 模式）表明步长方向错误或初始猜测远离解——应添加全局策略。运行 `globalization_advisor.py` （使用 `--far-from-solution` /报告失败情况），切换到信赖域或阻尼步长，而非放宽目标容差。
“信赖域步长降低了目标函数，所以接受并扩大半径。”	接受步长和扩大半径取决于收缩比ρ，而非仅符号。只有当ρ≥0.75且步长触达边界时， `step_quality.py` 才会标记 `expand` ；小正ρ（ `marginal` ）意味着接受步长但缩小半径。使用记录的 `trust_radius_action` 。
“Jacobian矩阵规模大且计算成本高，但全Newton方法是黄金标准，所以我还是要构建它。”	当n≥1000或Jacobian矩阵计算成本高时， `solver_selector.py` 会引导使用无矩阵Newton-Krylov（JFNK）方法，正是因为构建/分解J矩阵不可行；应使用Jacobian-向量乘积加预条件子替代。

Security

安全说明

Input Validation

输入验证

```
--size
```
(problem size) is validated as a positive integer, bounded at 10 billion
```
--residuals
```
are validated as finite non-negative numbers, capped at 100,000 entries
```
--tolerance
```
and
```
--target-tolerance
```
are validated as finite positive numbers
```
--problem-type
```
and
```
--constraint-type
```
are validated against fixed allowlists
```
--jacobian-quality
```
is validated against a fixed allowlist (
```
good
```
,
```
ill-conditioned
```
, etc.)

Step quality parameters (

predicted-reduction

actual-reduction

step-norm

gradient-norm

trust-radius

) are validated as finite numbers

```
--size
```
（问题规模）会被验证为正整数，上限为100亿
```
--residuals
```
会被验证为有限非负数，条目上限为10万条
```
--tolerance
```
和
```
--target-tolerance
```
会被验证为有限正数
```
--problem-type
```
和
```
--constraint-type
```
会与固定允许列表进行验证
```
--jacobian-quality
```
会与固定允许列表（
```
good
```
,
```
ill-conditioned
```
等）进行验证

步长质量参数（

predicted-reduction

actual-reduction

step-norm

gradient-norm

trust-radius

）会被验证为有限数

File Access

文件访问

```
jacobian_diagnostics.py
```
reads a single matrix file specified by
```
--matrix
```
; no directory traversal beyond the given path
Matrix files are size-limited and loaded with
```
allow_pickle=False
```
to prevent code execution
All other scripts read no external files; inputs are provided via CLI arguments
Scripts write only to stdout (JSON output)

```
jacobian_diagnostics.py
```
仅读取
```
--matrix
```
指定的单个矩阵文件；不会遍历给定路径之外的目录
矩阵文件有大小限制，加载时设置
```
allow_pickle=False
```
以防止代码执行
所有其他脚本不读取外部文件；输入通过CLI参数提供
脚本仅向标准输出（stdout）写入JSON结果

Tool Restrictions

工具限制

Read: Used to inspect script source, references, and user configuration files

Bash: Used to execute the six Python analysis scripts (

solver_selector.py

convergence_analyzer.py

jacobian_diagnostics.py

globalization_advisor.py

residual_monitor.py

step_quality.py

) with explicit argument lists

Write: Used to save analysis results or solver recommendations; writes are scoped to the user's working directory
Grep/Glob: Used to locate relevant files and search references

读取: 用于检查脚本源码、参考资料和用户配置文件

Bash: 用于执行6个Python分析脚本（

solver_selector.py

convergence_analyzer.py

jacobian_diagnostics.py

globalization_advisor.py

residual_monitor.py

step_quality.py

），并使用明确的参数列表

写入: 用于保存分析结果或求解器推荐；写入范围限定在用户工作目录
Grep/Glob: 用于定位相关文件和搜索参考资料

Safety Measures

安全措施

No
```
eval()
```
,
```
exec()
```
, or dynamic code generation
All subprocess calls use explicit argument lists (no
```
shell=True
```
)
Matrix dimension limits prevent memory exhaustion when loading Jacobian files
Residual history analysis operates on bounded-length numeric arrays only

不使用
```
eval()
```
、
```
exec()
```
或动态代码生成
所有子进程调用使用明确的参数列表（不使用
```
shell=True
```
）
矩阵维度限制可防止加载Jacobian文件时内存耗尽
残差历史分析仅对有限长度的数值数组进行操作

Limitations

局限性

No global convergence guarantee: All methods may fail for pathological problems
Jacobian accuracy: Finite-difference Jacobian may be inaccurate near discontinuities
Large dense problems: May require specialized solvers not covered here
Constrained optimization: Complex constraints need SQP or interior point methods

无全局收敛保证: 所有方法在面对病态问题时都可能失败
Jacobian矩阵精度: 有限差分Jacobian矩阵在不连续点附近可能不准确
大规模稠密问题: 可能需要此处未涵盖的专用求解器
约束优化: 复杂约束需要SQP或内点法

References

参考资料

```
references/solver_decision_tree.md
```
- Problem-based solver selection
```
references/method_catalog.md
```
- Method details and parameters
```
references/convergence_diagnostics.md
```
- Diagnosing convergence issues
```
references/globalization_strategies.md
```
- Line search and trust region

```
references/solver_decision_tree.md
```
- 基于问题的求解器选择指南
```
references/method_catalog.md
```
- 方法细节与参数说明
```
references/convergence_diagnostics.md
```
- 收敛问题诊断指南
```
references/globalization_strategies.md
```
- 线搜索与信赖域策略

Version History

版本历史

v1.2.2 (2026-06-24): Added a "Verification checklist" (evidence tied to each script's JSON outputs — convergence type/rate, residual patterns, Jacobian condition/finite-diff error, rank, trust-region step ratio, and solver/globalization agreement) and a "Common pitfalls & rationalizations" table covering constant-ratio-vs-superlinear, run-completion-vs-convergence, too-few-iterations, unverified analytic Jacobians, divergence handling, trust-region acceptance, and large/expensive-Jacobian routing
v1.2.0 (2026-06-23): Added
```
--problem-type
```
to
```
solver_selector.py
```
with a nonlinear least-squares path (Levenberg-Marquardt / Gauss-Newton); reordered solver selection so problem size dominates high-accuracy and routes large/expensive-Jacobian problems to Newton-Krylov; added
```
--far-from-solution
```
to
```
globalization_advisor.py
```
and surfaced Levenberg-Marquardt as the trust-region type for least-squares; corrected convergence classification so constant-ratio sequences are linear (not superlinear); RFC-8259-safe JSON (no
```
-Infinity
```
); input-validation hardening
v1.1.0 (2026-03-26): Optimized agent-discovery description, evaluation suite, security review docs, standardized metadata block, CHANGELOG
v1.0.0: Initial release with 6 analysis scripts

v1.2.2 (2026-06-24): 添加“验证检查清单”（与各脚本JSON输出绑定的证据——收敛类型/速率、残差模式、Jacobian矩阵条件数/有限差分误差、秩、信赖域步长比、求解器/全局策略一致性）和“常见陷阱与误区”表格，涵盖恒定比值vs超线性、运行完成vs收敛、迭代次数不足、未验证的解析Jacobian矩阵、发散处理、信赖域接受规则、大规模/高成本Jacobian矩阵路由等内容
v1.2.0 (2026-06-23): 为
```
solver_selector.py
```
添加
```
--problem-type
```
参数，支持非线性最小二乘路径（Levenberg-Marquardt / Gauss-Newton）；调整求解器选择逻辑，使问题规模优先于高精度需求，将大规模/高成本Jacobian矩阵问题引导至Newton-Krylov；为
```
globalization_advisor.py
```
添加
```
--far-from-solution
```
参数，并将Levenberg-Marquardt作为最小二乘问题的信赖域类型；修正收敛分类逻辑，将恒定比值序列标记为线性（而非超线性）；支持符合RFC-8259标准的JSON（无
```
-Infinity
```
）；强化输入验证
v1.1.0 (2026-03-26): 优化Agent发现描述、评估套件、安全审查文档，标准化元数据块，添加CHANGELOG
v1.0.0: 初始版本，包含6个分析脚本