曾聞小學堂的蟲食算考倒了大專生？想來其實也沒什麼奇怪！學而不思則罔，思其所學整理明辨，自有所得而且記憶長久，否則日遠時遷，早忘之矣！不會算豈不很正常！！舉例來說︰

孤獨的 $n$

在一堆空格的數式中，剛好只有一個數字 $n$ 。以下是一個「孤獨的六」：

$ \begin{matrix} & & & \Box & \Box & \Box \\ \hline \Box & ) & \Box & \Box & \Box & \Box \\ & & & \Box & & \\ \hline & & & \Box & \Box & \\ & & & & \Box & \\ \hline & & & & 6 & \Box \\ & & & & \Box & \Box \\ \end{matrix} $

這個孤獨的六是 $1053/9$ 的直式

求解需要個下手處。那個『被除數』的千位首數必是 1 。為什麼呢？觀其直式除法之『商』的首數與『除式寫法』未及於『被除數』的千位首數而得知。再者那個『除數』由『商』的尾數乘法知道，只能是『7 * 9 = 63』、『8 * 8 = 64』或『9 * 7 = 63』。添上直式除法首起『兩數相乘是一位數』，可以推知『商』的首數是 1 。否則就成兩位數的了。接著推『二位數減一位數等於一位數 1 』，此 1 來於其下的直式之寫法。如此可得『被除數』首起『二位數』為 10，而且『除數』是 9 。同理『商』的十位數也必是 1 ，以及 1 □ – 9 = 6 ，所以『被除數』的十位數是 5 。故得解。

『正算反推』雖都依賴『加減乘除』的『性質』，熟悉其一，不通另一，大半因為練習多寡所致。正如《邂逅 W!o ？！》文本中講的『逆問題』

Li Bai

A Quiet Night Thought

In front of my bed, there is bright moonlight.
It appears to be frost on the ground.
I lift my head and gaze at the August Moon,
I lower my head and think of my hometown.

Contemplation

Moon twilight approaches, coating the ground through the window,
Resembles a touch of frost,
Moon at the window,
Taking me back to where I am from.

李白

静夜思

床前明月光
疑是地上霜
舉頭望明月
低頭思故鄉

假使將李白的《靜夜思》翻譯成英文，藉由『中英對照』，是否更能『理解』原作之『意境』呢？還是會少了點『詩』的『味道』？？或許這個『利弊得失』就落在︰

『文化』之『盲點』，常顯現在『意義』的『忽略』之中。

『人文』之『偏見』，普遍藏於『字詞』之『情感』之內。

故而同一『內容』的多種『語言文本』，也許可見那『通常之所不見』。

【Inverse problem】

An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system. For example, if we have measurements of the Earth’s gravity field, then we might ask the question: “given the data that we have available, what can we say about the density distribution of the Earth in that area?” The solution to this problem (i.e., the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are some of the most important and well-studied mathematical problems in science and mathematics. Inverse problems arise in many branches of science and mathematics, including computer vision, natural language processing, machine learning, statistics, statistical inference, geophysics, medical imaging (such as computed axial tomography and EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.

【逆問題】

逆問題是一個關於如何將觀測和測量的結果轉換為物體或系統的信息的廣義框架。比如，如果我們有一個關於地球重力場的測量結果，我們就會問：「利用現有的信息，我們能否得到地球的密度分布？」。這類問題的解（即最符合測量數據的密度分布）通常就可以告訴我們一個無法直接測量的物理量。因此，逆問題是在數學和物理學中最重要和被研究的最多的問題之一。逆問題廣泛的出現在諸如計算機視覺，自然語言處理，機器學習，統計學，推論統計學，地理，醫學成像（比如X射線計算機斷層成像和腦電圖/事件相關電位），遙感，海洋聲學層析，無損檢測，航空，物理學中。

一般所以難解之故。所謂『去除盲點』，就是『能見己所未見』之功夫。

─── 《勇闖新世界︰《 PYDATALOG 》【專題】之約束編程‧五》

有人說︰『學問』就是要『學』要『問』。這裡彷彿假設『教』者存在，那麼『教』者是誰呢？所謂『三人行，必有我師焉』，明講『擇其善者而從之，其不善者而改之』，直指以『天下人』為『師』也！真能自『問』自『學』，又自『改』之，何道不通乎？？就像『未來』根源於『現在』，卻不決定於『過去』！！

故耳心思『邂逅 W!o ？！』，遙想『後裔之境』耶！？

如果一個『線性系統』 $S$ 的『輸出』 $O$ 與『輸入』 $I$ 間之『刺激響應』聯繫可以用『關係矩陣』 $M$ 表示為︰

$O = M \cdot I$

，那麼假使我們知道了 $O$ ，是否可以推測出 $I$ 是什麼？符號上或許可以形式的寫成︰

$I = M^{-1} \cdot O$

，要是『關係矩陣』之『逆矩陣』 $M^{-1}$ 存在的話。

由於可能的『非線性』，以及不同的『輸入組合』可以產生『相同輸出』，想要透過 $O$ 得到 $I$ ，一般是非常困難的。即使是在『線性系統』中，通常那個 $M^{-1}$ 也不存在！因此也許只能用著『最小平方法』 Least Squares 去推估那個系統 $S$ 之『最佳模型』的『組構參數』罷了！！

神經成像 Neuroimaging

『神經成像』 Neuroimaging 泛指能夠『直接』或『間接』的對神經系統 ── 主要是腦 ── 的『功能』，『結構』，以及『藥理』特性進行成像之技術。這個技術是現今之醫學，神經科學，和心理學較前沿的一個領域。

那麼我們可以依靠『圖像』來『解讀』一個人的『思想』、『情感』或是『健康』…等等的嗎？這樣的『讀解』和中醫『把脈』斷病如何比較其不同呢？？說不定神奇的『大自然』充滿著『象』與『相』，正等待心中有『數』的人去『解』的哩！！

或可倒過來寫『逆問題』解法發展史哩！☆

GEOPHYSICS

Jose Pujol (2007). ”The solution of nonlinear inverse problems and the Levenberg-Marquardt method.”GEOPHYSICS, 72(4), W1-W16.

https://doi.org/10.1190/1.2732552

TECHNICAL PAPERS

The solution of nonlinear inverse problems and the Levenberg-Marquardt method

Jose Pujol¹

¹University of Memphis, Department of Earth Sciences, Memphis, Tennessee. E-mail: jpujol@memphis.edu.

Although the Levenberg-Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems, its theoretical basis has not been described adequately in the literature. This is unfortunate, because Levenberg and Marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method. The idea of damping the solution was introduced by Levenberg, who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively. This result is not obvious, although it is taken for granted. Moreover, Levenberg derived a solution more general than the one currently used. Marquardt started with the current equation and showed that it interpolates between the ordinary least-squares-method and the steepest-descent method. In this tutorial, the two papers are combined into a unified presentation, which will help the reader gain a better understanding of what happens when solving nonlinear problems. Because the damped least-squares and steepest-descent methods are intimately related, the latter is also discussed, in particular in its relation to the gradient. When the inversion parameters have the same dimensions (and units), the direction of steepest descent is equal to the direction of minus the gradient. In other cases, it is necessary to introduce a metric (i.e., a definition of distance) in the parameter space to establish a relation between the two directions. Although neither Levenberg nor Marquardt discussed these matters, their results imply the introduction of a metric. Some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth. Finally, the work done by early researchers that rediscovered the damped least-squares method is put into a historical context.

重新論述『曲線擬合』函數選擇術矣！★

Curve fitting

Curve fitting^[1]^[2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points,^[3] possibly subject to constraints.^[4]^[5] Curve fitting can involve either interpolation,^[6]^[7] where an exact fit to the data is required, or smoothing,^[8]^[9] in which a “smooth” function is constructed that approximately fits the data. A related topic is regression analysis,^[10]^[11] which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization,^[12]^[13] to infer values of a function where no data are available,^[14] and to summarize the relationships among two or more variables.^[15] Extrapolation refers to the use of a fitted curve beyond the range of the observed data,^[16]and is subject to a degree of uncertainty^[17] since it may reflect the method used to construct the curve as much as it reflects the observed data.

Different types of curve fitting

Fitting functions to data points

Most commonly, one fits a function of the form $y = f (x)$ .

Fitting lines and polynomial functions to data points

Polynomial curves fitting points generated with a sine function.
Red line is a first degree polynomial, green line issecond degree, orange line is third degree and blue is fourth degree

The first degree polynomial equation

$\displaystyle y=ax+b\;$

is a line with slope a. A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates.

If the order of the equation is increased to a second degree polynomial, the following results:

$\displaystyle y=ax^{2}+bx+c\;.$

This will exactly fit a simple curve to three points.

If the order of the equation is increased to a third degree polynomial, the following is obtained:

$\displaystyle y=ax^{3}+bx^{2}+cx+d\;.$

This will exactly fit four points.

A more general statement would be to say it will exactly fit four constraints. Each constraint can be a point, angle, or curvature (which is the reciprocal of the radius of an osculating circle). Angle and curvature constraints are most often added to the ends of a curve, and in such cases are called end conditions. Identical end conditions are frequently used to ensure a smooth transition between polynomial curves contained within a single spline. Higher-order constraints, such as “the change in the rate of curvature”, could also be added. This, for example, would be useful in highway cloverleaf design to understand the rate of change of the forces applied to a car (see jerk), as it follows the cloverleaf, and to set reasonable speed limits, accordingly.

The first degree polynomial equation could also be an exact fit for a single point and an angle while the third degree polynomial equation could also be an exact fit for two points, an angle constraint, and a curvature constraint. Many other combinations of constraints are possible for these and for higher order polynomial equations.

If there are more than n + 1 constraints (n being the degree of the polynomial), the polynomial curve can still be run through those constraints. An exact fit to all constraints is not certain (but might happen, for example, in the case of a first degree polynomial exactly fitting three collinear points). In general, however, some method is then needed to evaluate each approximation. The least squares method is one way to compare the deviations.

There are several reasons given to get an approximate fit when it is possible to simply increase the degree of the polynomial equation and get an exact match.:

Even if an exact match exists, it does not necessarily follow that it can be readily discovered. Depending on the algorithm used there may be a divergent case, where the exact fit cannot be calculated, or it might take too much computer time to find the solution. This situation might require an approximate solution.
The effect of averaging out questionable data points in a sample, rather than distorting the curve to fit them exactly, may be desirable.
Runge’s phenomenon: high order polynomials can be highly oscillatory. If a curve runs through two points A and B, it would be expected that the curve would run somewhat near the midpoint of A and B, as well. This may not happen with high-order polynomial curves; they may even have values that are very large in positive or negative magnitude. With low-order polynomials, the curve is more likely to fall near the midpoint (it’s even guaranteed to exactly run through the midpoint on a first degree polynomial).
Low-order polynomials tend to be smooth and high order polynomial curves tend to be “lumpy”. To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to negative. We can also say this is where it transitions from “holding water” to “shedding water”. Note that it is only “possible” that high order polynomials will be lumpy; they could also be smooth, but there is no guarantee of this, unlike with low order polynomial curves. A fifteenth degree polynomial could have, at most, thirteen inflection points, but could also have twelve, eleven, or any number down to zero.

The degree of the polynomial curve being higher than needed for an exact fit is undesirable for all the reasons listed previously for high order polynomials, but also leads to a case where there are an infinite number of solutions. For example, a first degree polynomial (a line) constrained by only a single point, instead of the usual two, would give an infinite number of solutions. This brings up the problem of how to compare and choose just one solution, which can be a problem for software and for humans, as well. For this reason, it is usually best to choose as low a degree as possible for an exact match on all constraints, and perhaps an even lower degree, if an approximate fit is acceptable.

※ 註

$e^{kx}$ 函數無窮可微 $\frac{d^n}{{dx}^n} e^{kx} = k^n e^{kx}$ ，無限平滑。

$\cos (\theta) = \frac{1}{2} \left( e^{i \theta} + e^{- i \theta} \right)$

$\sin (\theta) = \frac{1}{2i} \left( e^{i \theta} - e^{- i \theta} \right)$

進而跨越『混沌之澤』呦◎

Rössler attractor

The Rössler attractor /ˈrɒslər/ is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler.^[1]^[2] These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor.^[3]

Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams. The original Rössler paper states the Rössler attractor was intended to behave similarly to the Lorenz attractor, but also be easier to analyze qualitatively.^[1] An orbit within the attractor follows an outward spiral close to the $\displaystyle x,y$ plane around an unstable fixed point. Once the graph spirals out enough, a second fixed point influences the graph, causing a rise and twist in the $\displaystyle z$ -dimension. In the time domain, it becomes apparent that although each variable is oscillating within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the Lorenz attractor, but is simpler and has only one manifold. Otto Rössler designed the Rössler attractor in 1976,^[1] but the originally theoretical equations were later found to be useful in modeling equilibrium in chemical reactions.

Rössler attractor as a stereogramwith $\displaystyle a=0.2$ , $\displaystyle b=0.2$ , $\displaystyle c=14$

Definition

The defining equations of the Rössler system are:^[3]

$\displaystyle {\begin{cases}{\frac {dx}{dt}}=-y-z\\{\frac {dy}{dt}}=x+ay\\{\frac {dz}{dt}}=b+z(x-c)\end{cases}}$

Rössler studied the chaotic attractor with $\displaystyle a=0.2$ , $\displaystyle b=0.2$ , though properties of $\displaystyle a=0.1$ , $\displaystyle b=0.1$ , and $\displaystyle c=14$ have been more commonly used since. Another line of the parameter space was investigated using the topological analysis. It corresponds to $\displaystyle b=2$ , $\displaystyle c=4$ , and $\displaystyle a$ was chosen as the bifurcation parameter.^[4] How Rössler discovered this set of equations was investigated by Letellier and Messager.^[5]