1.Introduction SaknarinChannark ,PoomKumam ,ParinChaipunya ,WachirapongJirakitpuwapat ProgressiveIterativeApproximationMethodwithMemoryandSequencesofWeightsforLeastSquareCurveFitting

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Vol.28 No.1 January - March 2023 Page: [90-107]

Original research article

Progressive Iterative Approximation Method with Memory and Sequences of Weights for Least Square Curve Fitting

Saknarin Channark^1,2, Poom Kumam^1,2,3*, Parin Chaipunya^1,2, Wachirapong Jirakitpuwapat⁴

1Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand

2Center of Excellence in Theoretical and Computational Science, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi, Bangkok 10140, Thailand

3Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

4National Security and Dual-Use Technology Center, National Science and Technology Development Agency, Pathum Thani 12120, Thailand

Received 1 November 2021; Received in revised form - Accepted 3 August 2022; Available online 20 March 2023

ABSTRACT

The progressive iterative approximation method with memory and sequences of weights for least square curve fitting (SSLSPIA) is presented in this paper. This method improves the MLSPIA method by varying the weights of the moving average between it- erations, using three sequences of weights derived from the singular values of a colloca- tion matrix. It is proved that a sequence of fitting curves with an appropriate alternative of weights converge to the solution of least square fitting and that the convergence rate of the new method is faster than that of the MLSPIA method. Some examples and applications in this paper prove the SSLSPIA method is superior.

Keywords: Least square curve fitting; Progressive iterative approximation; Sequences of weights

1. Introduction

Least-squares fitting (LSF) to data points is defined using a parametric curve which is a commonly used method in sci-

entific and engineering research, involving geometric modeling and Computer Aided Design (CAD) [1–3]. It is widely used in- side numerous applications. Due to its ra-

pidity, it’s especially useful when dealing with enormous data sets. LSF can be di- vided into two types which are derived from several metric descriptions. Control points and parameter knots are variables in one, and the problems are addressed using non- linear least squares. Due to the nonlinear nature of them, a sufficient initial fitting is necessary. The second is data collection that has been organized by giving each data point a parameter. The control points can then be obtained by solving the linear LSF.

This may be a simpler construction method because it is linear. Details and examples are provided in [1, 2, 4–21] and references therein.

Smoothness and evenness are both enhanced with the fitting curves and sur- faces, according to user-defined parameters (occasionally regarded, [2,5,6,12,22]). Al- though there is no ideal alternative to these user-defined settings, the smoothness and precision of the fits may be impacted. Re- searchers devised a method for defining a relatively small amount of control points [9] in order to improve fitting efficiency.

When examining linear LSF within a cer- tain amount of control points, a value of pa- rameters comes from each point of the data set, we will assume no fairness terms in this article. Standard fits to each type of data collection can obtain the control points by directly resolving consistent systems of lin- ear equations. It is possible to collect and solve a new type of linear equation.

One of the LSF methods that solves a linear system is the MLSPIA method, which is a least square form of the PIA method [23] and LSPIA method [8] cre- ated directly in [24]. It constructs a se- quence of converging curves to arrive at a least square fitting solution by repeatedly approximating control points and construct- ing a sequence of converging curves. It is

a straightforward approach to obtain a suit- able solution that maintains the properties by maintaining the shape. When an extra control point is introduced in [8, 13, 24], just the iterative formula has to be changed rather than developing a new linear system.

In this work, Algorithm 1 presents a progressive iterative approximation method with memory and sequences of weights for least square curve fitting (SSLSPIA) to improve convergence rate. The method frequently creates a sequence of curves by iteratively updating the control points and the sequence’s weighted sums. The justification for naming this new tech- nique ”SSLSPIA” is detailed in section 3.

The following are our contributions to the SSLSPIA method:

• Whether or not the collocation ma- trix has a deficient column rank, the SSLSPIA method is feasible.

• The SSLSPIA method has a higher convergence rate than the MLSPIA method when the collocation matrix has full column rank [24].

• The SSLSPIA method conserves the MLSPIA benefits, including the abil- ity to choose a large amount of set points, and allowing the number of starting control points, the shape con- serving property, and parallel execu- tion.

• The SSLSPIA method uses semi- constant knots for fitting curve which differs from knots in the MLSPIA method.

• The MLSPIA method is a variant of the SSLSPIA method that uses con- stant weight sequences.

The following is how the article is organized. In Section 2, there is a brief overview of related works. Section 3 pro- poses an iterative format and a convergence analysis for the SSLSPIA method. Section 4 performs the implementation, which in- cludes three examples, and Section 5 de- scribes the SSLSPIA applications. Finally, Section 6 expresses the conclusion.

2. Related work

Progressive Iterative Approximation (PIA) methods feature iterative forms simi- lar to geometric interpolation (GI) methods, as illustrated in [25, 26], and other exam- ples. The PIA methods rely on parametric distance identically to the GI methods. The initial discoveries were made in the 1970s by de Boor [27, 28], and Yamaguji [29] . Each version of the PIA method creates a sequence of curves by adjusting parameters and directly solving a linear system.

Assume that {𝑄𝑗}^𝑚_𝑗=1 is an ordered point set for fitted and {0 = 𝑡1 < 𝑡2 <

𝑡3 < · · · < 𝑡_𝑚 = 1} is an increasing se- quence parameters of {𝑄𝑗}^𝑚_𝑗=1. At the be- ginning of the iteration, we select {𝑃_𝑖⁰}_𝑖^𝑛₌₁ from {𝑄_𝑗}^𝑚_𝑗=1 to be the control point set and generate a segment of blending curve 𝐶⁰(𝑡),i.e.,

𝐶⁰(𝑡) = Õ𝑛

𝑖=1

𝐵𝑖(𝑡)𝑃⁰_𝑖, 𝑡∈ [𝑡1, 𝑡𝑚], (2.1) where {𝐵𝑖(𝑡)}^𝑛_𝑖=1 is a normalized totally positive (NTP) that is used as the blending base for real functions.

We know that if 𝐵_𝑖(𝑡) ≥ 0, 𝑖 ∈ {1,2,3, . . . , 𝑛}andÍ_𝑛

𝑖=1𝐵_𝑖(𝑡) = 1, the ba- sis is NTP. For each𝑡 in[0,1], hold. Any non-decreasing series of collocation matrix is a totally positive matrix, meaning that ev- ery one of the minors in the matrix is non- negative [30, 31]. On0 = 𝑡1 < 𝑡2 < 𝑡3 <

· · · < 𝑡𝑚 =1, the collocation matrix of the

NTP blending basis is

𝐵𝑚×𝑛=©­

­­­

«

𝐵1(𝑡1) 𝐵2(𝑡1) ... 𝐵𝑛(𝑡1) 𝐵1(𝑡2) 𝐵2(𝑡2) ... 𝐵𝑛(𝑡2) . .. . .. ... . ..

𝐵1(𝑡𝑚) 𝐵2(𝑡𝑚) ... 𝐵𝑛(𝑡𝑚) ª®®®

®

¬ , (2.2)

With a point set{𝑄_𝑗}^𝑚_𝑗=1and starting control point set {𝑃⁰_𝑖}^𝑛_𝑖=1, the PIA method assigned in [23], the LSPIA method in [8]

and the MLSPIA method in [24] approxi- mate the curve with the NTP basis by the following curves repetitively:

𝐶^𝑘+1(𝑡)= Õ𝑛 𝑖=1

𝐵𝑖(𝑡)𝑃_𝑖^𝑘+1, 𝑡∈ [0,1], 𝑘≥0, (2.3)

the control points are adjusted

𝑃_𝑖^𝑘+1=𝑃_𝑖^𝑘+Δ^𝑘_𝑖, 𝑖∈ {1,2,3, . . . , 𝑛}, 𝑘≥0, (2.4)

where adjusting the vector

Δ_𝑖^𝑘=

(𝑄𝑖−𝐶^𝑘(𝑡𝑖), for PIA method, 𝜇Í_𝑚

𝑗=1𝐵𝑖(𝑡𝑗) (𝑄𝑗−𝐶^𝑘(𝑡𝑗)),for LSPIA method,

𝜇is a constant, 𝑖 ∈ {1,2,3, . . . , 𝑛}, 𝑘 ≥ 0, and adjusting the vector for MLSPIA method





Δ⁰_𝑖 =𝜈Í_𝑚

𝑗=1𝐵𝑖(𝑡𝑗)Φ⁰_𝑗,

Δ^𝑘_𝑖 =(1−𝜔)Δ_𝑖^𝑘−1+𝛾 𝛿_𝑖^𝑘+ (𝜔−𝛾)𝛿_𝑖^𝑘−1, 𝑘≥1, 𝛿_𝑖^𝑘=𝜈Í_𝑚

𝑗=1𝐵𝑖(𝑡𝑗) (𝑄𝑗−𝐶^𝑘(𝑡𝑗)), 𝑘≥0,

byΦ⁰_𝑗is a vector where{𝑄𝑗}^𝑚_𝑗=1is located for every 𝑗 ∈ {1,2,3, . . . , 𝑚}, and 𝜔, 𝛾, 𝜈 are real weights.

The PIA method in the format of Eqs.

(2.3)-(2.4), described and originally labeled in [32], depends on the notion of benefit and loss improvement as proposed in [27, 28]., and is based on the non-uniform B-spline foundation. Of [33], the PIA approach in [23] evolved into a weighted PIA method, which accelerates convergence.

In the conventional PIA format, how- ever, the amount of control points is equal to the amount of input points. This is not feasi- ble when the amount of input points is really large. A new extended PIA (EPIA) format

was introduced in [34], in which the amount of control points is less than the amount of input points. Because of its local property and parallel processing capabilities, the ex- tended PIA is an ideal approach for fitting an extensive amount of data.

In addition, [35] has been expanded to approximate NURBS curves. According to the local property of the PIA method [36], if a subset of the control points is repeat- edly updated, only the limit curve interpo- lates the subset of input points, leaving the others unchanged. The PIA methods con- verge with appropriate weights and have the properties of convexity conserving, as well as the evident exposition of curves, locality, and adaptivity. By altering only the control points which affect this segment to lower- cost [23, 32], the locality and adaptivity en- sure improved probability that the resultant curve segment approximation is accurate.

More details are shown in [13].

The LSPIA method assigned in [8]

is primarily designed for large sets of data and inherits the benefits of PIA methods in the form of Eqs. (2.3)-(2.4). The LSPIA method has two distinct advantages. For starters, the LSPIA method can fit much larger sets quickly and reliably. Second, when using the LSPIA method for incre- mental data fitting, a new set of repetitively may be begun from the initial result of the previous output, saving a significant pro- cessing. T-splines in [7] are used to con- struct it. Then, in [10], replace the B-spline basis with the generalized B-spline basis to get the weighted LSF curve. In the nonsin- gular situation, the LSPIA method [8] con- verges, whereas [11] proves convergence in the singular case.

The PIA method with memory for least square fitting (MLSPIA) is one (LSF) method which is ineffectual for solving lin- ear systems, a least square version of PIA

method [23], and the LSPIA method [8]

created directly in [24] to enhance conver- gence rate. It repeatedly estimates control points and creates a sequence of converging curves to arrive at a least square fitting solu- tion. It is a way to obtain a suitable solution that maintains the properties by maintaining the form.The construction of the MLSPIA method in the format of Eqs. (2.3)-(2.4), assigned in [8], is analogous to the LSPIA method in the format of Eqs. (2.3)-(2.4), as- signed in [8], except for the difference ofΔ_𝑖^𝑘 for each possible𝑖and𝑘, which is similar to the LSPIA method. For appropriate weights 𝜔, 𝜈, 𝛾, it also converged to the LSF curve of{𝑄𝑗}^𝑚_𝑗=1.

In this article, we will compare the MLSPIA method from the difference of three real weights to the difference of three real sequences of weights. It will con- verge to the LSF curve as well. Further- more, for the cubic B-spline fitting curve, we use semi-constant knots, which is differ- ent from knots used in the LSPIA and ML- SPIA methods.

3. The SSLSPIA method

We will present the SSLSPIA method, which is a progressive iterative approximation method for the LSF curve with memory and weight sequences, and analyze its convergence rate.

With the provided data point set {𝑄_𝑗}^𝑚_𝑗=1 and starting control point set {𝑃⁰_𝑖}^𝑛_𝑖=1, and 𝑚 > 𝑛, the NTP basis 𝐵1(𝑡), 𝐵2(𝑡), 𝐵3(𝑡), . . . , 𝐵𝑛(𝑡) and the se- quence0=𝑡1 < 𝑡2 < 𝑡3 <· · · < 𝑡_𝑚 =1, the PIA method [23], LSPIA method [8], and MLSPIA method [24] iteratively approxi- mate the curve by Eqs. (2.3)-(2.4) in the vector space where{𝑄_𝑗}^𝑚_𝑗=1is located.

The SSLSPIA method iteratively ap- proximates the curve using Eqs. (2.3)- (2.4) by resetting the control points with

the adjusting vector again from the previous curve.





Δ⁰_𝑖 =𝜈𝑘Í_𝑚

𝑗=1𝐵𝑖(𝑡𝑗)Φ⁰_𝑗,

Δ^𝑘_𝑖 =(1−𝜔𝑘)Δ_𝑖^𝑘−1+𝛾𝑘𝛿_𝑖^𝑘+ (𝜔𝑘−𝛾𝑘)𝛿_𝑖^𝑘−1, 𝑘 ≥1, 𝛿_𝑖^𝑘=𝜈𝑘Í_𝑚

𝑗=1𝐵𝑖(𝑡𝑗) (𝑄𝑗−𝐶^𝑘(𝑡𝑗)), 𝑘≥0, (3.1)

where Φ⁰_𝑗 is a point in the vector space which {𝑄_𝑗}^𝑚_𝑗=1 is located for every 𝑗 ∈ {1,2,3, . . . , 𝑚}, and𝜔_𝑘, 𝛾_𝑘, 𝜈_𝑘 are real se- quences of weights.

It is called the SSLSPIA method as its construction is expected to develop the difference of three real weight 𝜔,𝛾, 𝜈 to the difference of three real sequences of weights 𝜔_𝑘, 𝛾_𝑘, 𝜈_𝑘 for each possible 𝑘, which is similar to the MLSPIA method defined in [24]. It will also converge to the LSF curve of {𝑄_𝑗}^𝑚_𝑗=1 for suitable se- quences of weights 𝜔𝑘, 𝜈𝑘, 𝛾𝑘 which we can discuss later. It is a memory method because when we use Eq. (3.1) to com- pute Δ^𝑘−_𝑖 ¹, we have to gather and utilize Δ^𝑘−_𝑖 ¹and𝛿^𝑘−_𝑖 ¹from the preceding curve, as well as compute 𝛿^𝑘_𝑖 for every 𝑘 ≥ 1 and 𝑖∈ {1,2,3, . . . , 𝑛}. Furthermore, as shown in section 4, we use semi-constant knots for the cubic B-spline fitting curve, which differ from knots used in the LSPIA and MLSPIA methods. In [24], the SSLSPIA method has a higher convergence rate than MLSPIA.

The equivalent formulation of the SSLSPIA methods developed by Eqs.

(2.3)-(3.1) is first given. The proof follow- ing line by line of the proof of [24].

Lemma 3.1. Let {𝑃⁰_𝑖}^𝑛_𝑖=1 be an starting control point set and let{Φ⁰_𝑗}^𝑚_𝑗=1be a point set. The 𝑃_𝑖^𝑘, 𝑖 ∈ {1,2,3, . . . , 𝑛}, 𝑘 ≥ 0 constructed by Eqs. (2.4)-(3.1). They can be rewritten as

𝑃_𝑖^𝑘+1=𝑃_𝑖^𝑘+𝜈𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Φ^𝑘_𝑗, 𝑖∈ {1,2,3, . . . , 𝑛}, 𝑘 ≥0, (3.2)

where

Φ^𝑘+_𝑗 ¹=(1−𝜔𝑘)Φ^𝑘_𝑗+𝜔𝑘𝑄𝑗

−Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)

𝜔𝑘𝑃_𝑖^𝑘+𝛾𝑘𝜈𝑘 Õ𝑚 𝑗1=1

𝐵𝑖(𝑡𝑗1)Φ^𝑘_𝑗1

, 𝑘 ≥0. (3.3)

Proof. It is simple to affirm that Eq. (3.2) contains for 𝑘 = 0 via Eqs. (2.4)-(3.1).

Assume that for some 𝑘 ≥ 0, Eq. (3.2) holds true. As a consequence, for each𝑖 ∈ {1,2,3, . . . , 𝑛},

𝜔𝑘𝛿_𝑖^𝑘+𝛾𝑘(𝛿_𝑖^𝑘+1−𝛿_𝑖^𝑘)

=𝜈𝑘𝜔𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)

𝑄𝑗−𝐶^𝑘(𝑡𝑗)

−𝜈𝑘𝛾𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)

𝐶^𝑘+1(𝑡𝑗) −𝐶^𝑘(𝑡𝑗)

=𝜈𝑘𝜔𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗) 𝑄𝑗−Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)𝑃_𝑖^𝑘

!

−𝜈𝑘𝛾𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)

𝑃_𝑖^𝑘+1−𝑃_𝑖^𝑘 ,

holds by Eq. (3.1), we get by Eqs. (2.4)- (3.1) for every𝑖∈ {1,2,3, . . . , 𝑛}that

Δ^𝑘+_𝑖 ¹=(1−𝜔𝑘)Δ_𝑖^𝑘+𝜔𝑘𝛿_𝑖^𝑘+𝛾𝑘(𝛿_𝑖^𝑘+1−𝛿_𝑖^𝑘)

=(1−𝜔𝑘)

𝑃_𝑖^𝑘+1−𝑃_𝑖^𝑘

+𝜈𝑘𝜔𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗) 𝑄𝑗− Õ𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)𝑃_𝑖^𝑘

!

−𝜈𝑘𝛾𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)

𝑃_𝑖^𝑘+1−𝑃_𝑖^𝑘

=(1−𝜔𝑘)𝜈𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Φ^𝑘_𝑗 +𝜈𝑘𝜔𝑘

Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗) 𝑄𝑗−Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗)𝑃_𝑖^𝑘

!

−𝜈_𝑘²𝛾𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑗) Õ^𝑛 𝑗¹=1

𝐵𝑖(𝑡𝑗¹)Φ^𝑘_𝑗1

=𝜈𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Φ^𝑘+_𝑗 ¹,

if we give

Φ^𝑘+_𝑗 ¹=(1−𝜔𝑘)Φ^𝑘_𝑗+𝜔𝑘𝑄𝑗

−Õ^𝑛 𝑖=1

𝐵𝑖(𝑡𝑖)

𝜔𝑘𝑃_𝑖^𝑘+𝛾𝑘𝜈𝑘 Õ𝑚 𝑗¹=1

𝐵𝑖(𝑡𝑗¹)Φ^𝑘_𝑗1



.

It follows that, for𝑖∈ {1,2,3, . . . , 𝑛},

𝑃_𝑖^𝑘+2=𝑃_𝑖^𝑘+1+Δ^𝑘+_𝑖 ¹=𝑃_𝑖^𝑘+1+𝜈𝑘 Õ𝑚 𝑗=1

𝐵𝑖(𝑡𝑗)Φ^𝑘+_𝑗 ¹,

which means that Eq. (3.2) holds for 𝑘 + 1. From mathematical induction, Eq. (3.2)

holds for every𝑘 ≥ 0. □

Denote

( Φ_𝑖^𝑘=

Φ^𝑘₁,Φ^𝑘₂,Φ^𝑘₃, . . . ,Φ^𝑘_𝑚𝑇

, 𝑘 ≥0, 𝑃_𝑖^𝑘=

𝑃1^𝑘, 𝑃2^𝑘, 𝑃3^𝑘, . . . , 𝑃𝑛^𝑘𝑇 (3.4)

and

𝑄𝑖=[𝑄1, 𝑄2, 𝑄3, . . . , 𝑄𝑚]^𝑇 , 𝐵= 𝐵𝑖(𝑡𝑗) 𝑚×𝑛,

(3.5)

where𝑃_𝑖^𝑘, 𝑖 ∈ {1,2,3, . . . , 𝑛},andΦ^𝑘_𝑗, 𝑗 ∈ {1,2,3, . . . , 𝑚},are assigned by Eqs. (3.2)- (3.3), respectively, for every𝑘 ≥ 0.There- fore Eqs. (3.2)-(3.3) can be written as

( Φ^𝑘+1=

(1−𝜔𝑘)𝐼𝑚−𝛾𝑘𝜈𝑘𝐵𝐵^𝑇𝑇

Φ^𝑘+𝜔𝑘(𝑄−𝐵𝑃^𝑘), 𝑃^𝑘+1=𝜈𝑘𝐵^𝑇Φ^𝑘+𝑃^𝑘, 𝑘≥0,

(3.6)

or equivalent to,

Φ^𝑘+1 𝑃^𝑘+1

=𝐻𝜔_𝑘,𝛾_𝑘,𝜈_𝑘 Φ^𝑘

𝑃^𝑘

+𝐶𝜔_𝑘,𝛾_𝑘,𝜈_𝑘, 𝑘 ≥0, (3.7)

where

𝐻𝜔_𝑘,𝛾_𝑘,𝜈_𝑘=

(1−𝜔𝑘)𝐼𝑚−𝛾𝑘𝜈𝑘𝐵𝐵^𝑇 −𝜔𝑘𝐵

𝜈𝑘𝐵^𝑇 𝐼𝑛

, (3.8)

and

𝐶𝜔_𝑘,𝛾_𝑘,𝜈_𝑘= 𝜔𝑘𝑄

0

.

The matrix iterative format can be thought of as Eq. (3.7). As a result, we an- alyze the convergence of

Φ^𝑘 𝑃^𝑘

provided by Eq. (3.7) from

Φ⁰ 𝑃⁰

while considering the curves generated by the SSLSPIA method.

Assumption 1. Assume that Eq. (3.5) as- signs𝑚 > 𝑛and𝐵. The singular values of the matrix 𝐵are 𝑟𝑎𝑛𝑘(𝐵) = 𝑟 𝑎𝑛𝑑 𝜎1 ≥ 𝜎2 ≥𝜎3 ≥ . . .≥ 𝜎𝑟 >0. Suppose that

0< 𝜔𝑘 <2, 𝜔𝑘− 𝜔𝑘

𝜎1²𝜈𝑘 < 𝛾𝑘< 𝜔𝑘

2 −𝜔𝑘−2 𝜎1²𝜈𝑘 , 𝜈𝑘>0.

(3.9)

Lemma 3.2. Under the Assumption 1, sup- pose that Í

𝑟 = 𝑑𝑖𝑎𝑔(𝜎1, 𝜎2, 𝜎2, . . . , 𝜎𝑟) with 𝜎1 ≥ 𝜎2 ≥ 𝜎3 ≥ . . . ≥ 𝜎_𝑟 > 0. 𝜆 is an eigenvalue of

𝐻ˆ𝜔_𝑘,𝛾_𝑘,𝜈_𝑘

=©­

«

(1−𝜔𝑘)𝐼𝑟−𝛾𝑘𝜈𝑘Í2

𝑟 0 −𝜔𝑘Í

𝑟 0 (1−𝜔𝑘)𝐼𝑚−𝑟 0 𝜈𝑘Í

𝑟 0 𝐼𝑟

ª®

¬ , (3.10)

equivalent to𝜆=1−𝜔_𝑘 or𝜆is a solution of the one of the equations as follows

𝜆²+

𝛾𝑘𝜈𝑘𝜎_𝑖²− (2−𝜔𝑘)

𝜆+𝜎_𝑖²𝜈𝑘(𝜔𝑘−𝛾𝑘)+1−𝜔𝑘=0, (3.11)

where 𝑖∈ {1,2,3, . . . , 𝑟}.

Proof. Because the matrices𝐻𝜔𝑘,𝛾𝑘,𝜈𝑘 and 𝐻ˆ_{𝜔𝑘,𝛾𝑘,𝜈𝑘} are orthogonally identical, their eigenvalues are the same. We consider that

|𝐻ˆ𝜔_𝑘,𝛾_𝑘,𝜈_𝑘−𝜆𝐼𝑚+𝑟|

=

(1−𝜔𝑘−𝜆)𝐼𝑟−𝛾𝑘𝜈𝑘Í2

𝑟 0 −𝜔𝑘Í

𝑟 0 (1−𝜔𝑘−𝜆)𝐼𝑚−𝑟 0 𝜈𝑘Í

𝑟 0 (1−𝜆)𝐼𝑟

=(1−𝜔𝑘−𝜆)^𝑚−𝑟

×Ö^𝑟 𝑖=1

(1−𝜆) (1−𝜔𝑘−𝜆−𝛾𝑘𝜈𝑘𝜎_𝑖²) +𝜔𝑘𝜈𝑘𝜎_𝑖² .

So𝜆is an eigenvalue of𝐻ˆ_{𝜔𝑘,𝛾𝑘,𝜈𝑘} equiva- lent to𝜆=1−𝜔_𝑘 or

Ö𝑟 𝑖=1

(1−𝜆) (1−𝜔𝑘−𝜆−𝛾𝑘𝜈𝑘𝜎_𝑖²) +𝜔𝑘𝜈𝑘𝜎²_𝑖

=0.

Therefore,𝜆=1−𝜔𝑘 or𝜆is a solution of the equations Eq. (3.11). □ Lemma 3.3. [37] The solutions of the real quadratic equation 𝜆² − 𝑏𝜆 + 𝑐 = 0 are smaller than unity in modulus, equivalent to

|𝑐| <1and|𝑏| < 1+𝑐.

Lemma 3.4. Under the Assumption 1, 𝜌(𝐻ˆ_{𝜔𝑘,𝛾𝑘,𝜈𝑘}) < 1, where𝜌(𝐻ˆ_{𝜔𝑘,𝛾𝑘,𝜈𝑘}) de- notes the spectral radius of 𝐻_{𝜔𝑘,𝛾𝑘,𝜈𝑘} as- signed by Eq.(3.10), equivalent to weights 𝜔_𝑘, 𝛾_𝑘, 𝜈_𝑘 satisfy Eq.(3.9).

Proof. Defined weights 𝜔_𝑘, 𝛾_𝑘, 𝜈_𝑘, by Lemma 3.3 unity in modulus are larger than all the solutions of equations in Eq. (3.11) if and only if

(|𝜎_𝑖²𝜈𝑘(𝜔𝑘−𝛾𝑘) +1−𝜔𝑘|<1, 𝑖∈ {1,2,3, . . . , 𝑟},

|𝛾𝑘𝜈𝑘𝜎_𝑖²− (2−𝜔𝑘) |<2−𝜔𝑘+𝜎²_𝑖𝜈𝑘(𝜔𝑘−𝛾𝑘).

(3.12)

Then, by Lemma 3.2𝜌(𝐻ˆ_{𝜔𝑘,𝛾𝑘,𝜈𝑘}) < 1if and only if|𝜔_𝑘−1| <1and Eq. (3.11) holds.

Or equivalently, equations in Eq. (3.11) are smaller than unity in modulus if and only if





0< 𝜔𝑘<2, 𝜎_𝑖²𝜈𝑘𝜔𝑘>0,

𝜔𝑘−2< 𝜎_𝑖²𝜈𝑘(𝜔𝑘−𝛾𝑘)< 𝜔𝑘, 𝜎_𝑖²𝜈𝑘(𝜔𝑘−2𝛾𝑘)>2(𝜔𝑘−2),

𝑖∈ {1,2,3, . . . , 𝑟}.

(3.13)

Eq. (3.9) is equivalent to Eq. (3.13) because 𝜎₁² ≥ 𝜎₂² ≥𝜎₃² ≥ . . .≥ 𝜎_𝑟² > 0. □ Theorem 3.5. Assume that the Assump- tion 1. Then, for every starting con- trol point set {𝑃⁰_𝑖}^𝑛_𝑖=1 and every point set {Φ⁰_𝑗}^𝑚_𝑗=1, the curve sequences which are created by the SSLSPIA method, assigned by Eq. (2.3), Eq. (2.4) and Eq. (3.1), with weights 𝜔𝑘, 𝛾𝑘, 𝜈𝑘, converge to the LSF curve of the given{𝑄⁰_𝑗}^𝑚_𝑗=1.

Proof. The rewritten format of the iteration matrix𝐻_{𝜔𝑘,𝛾𝑘,𝜈𝑘}assigned by Eq. (3.8). As- sume that the singular value decomposition (SVD) of𝐵is

𝐵=𝑈Í

𝑟 0

0 0

𝑉^𝑇, (3.14) for 𝑈 ∈ R^𝑚×𝑚 and 𝑉 ∈ R^𝑛×𝑛 are the orthogonal matrices and Í

𝑟 =

𝑑𝑖𝑎𝑔(𝜎1, 𝜎2, 𝜎3, . . . , 𝜎_𝑟) with 𝜎1 ≥ 𝜎2 ≥ 𝜎3 ≥. . .≥ 𝜎_𝑟 > 0. Suppose

𝑊 = 𝑈 0

0 𝑉

, (3.15)

then𝑊 ∈ R^{(𝑚+𝑛)×(𝑚+𝑛)} is the orthogonal matrix. Next,

𝑈^𝑇𝐵𝑉 =

Í𝑟 0 0 0

∈R^𝑚×𝑛, 𝑉^𝑇𝐵^𝑇𝑈=

Í𝑟 0 0 0

∈R^𝑛×𝑚, and

𝑈^𝑇𝐵𝐵^𝑇𝑈= Í₂

𝑟 0

0 0

∈R^𝑚×𝑚, it holds that

𝑊^𝑇𝐻𝜔_𝑘,𝛾_𝑘,𝜈_𝑘𝑊 =

𝐻ˆ𝜔_𝑘,𝛾_𝑘,𝜈_𝑘 0 0 𝐼𝑛−𝑟

, (3.16)

where

𝐻ˆ𝜔_𝑘,𝛾_𝑘,𝜈_𝑘

=©­

«

(1−𝜔𝑘)𝐼𝑟−𝛾𝑘𝜈𝑘Í2

𝑟 0 −𝜔𝑘Í

𝑟 0 (1−𝜔𝑘)𝐼𝑚−𝑟 0 𝜈𝑘Í

𝑟 0 𝐼𝑟

ª®

¬ . (3.17)

We focus on the convergence of the sequence {𝑃^𝑘}^∞_𝑘=0 because proving that the curve sequence {𝐶^𝑘(𝑡)}^∞_𝑘=0 generated by the SSLSPIA method converges to a LSF curve is equivalent to proving that the repetitively sequence {𝑃^𝑘}^∞_𝑘=0 created by Eq. (3.6) converges to a solution of 𝐵^𝑇𝐵𝑋 =𝐵^𝑇𝑄,where𝐵and𝑄are assigned by Eq. (3.5). For any weights𝜔_𝑘, 𝛾_𝑘, 𝜈_𝑘 satisfying Eq. (3.9), let𝛼=𝑄−𝐵𝛽, where 𝛽is the solution of 𝐵^𝑇𝐵𝑋 = 𝐵^𝑇𝑄, then it can be simply affirmed that(𝛼^𝑇𝛽^𝑇)^𝑇 is the result of the equation

(𝐼𝑚+𝑛−𝐻𝜔_𝑘,𝛾_𝑘,𝜈_𝑘) (𝑥^𝑇𝑦^𝑇)^𝑇 =𝐶𝜔_𝑘,𝛾_𝑘,𝜈_𝑘, (3.18)

where 𝐻_{𝜔𝑘,𝛾𝑘,𝜈𝑘} and 𝐶_{𝜔𝑘,𝛾𝑘,𝜈𝑘}, are as- signed in Eq. (3.8). That means, the equa- tion Eq. (3.18) is compatible. Assume

Φ 𝑃

be the result of the equation Eq. (3.18), i.e.,

(𝐼𝑚+𝑛−𝐻𝜔_𝑘,𝛾_𝑘,𝜈_𝑘) Φ

𝑃

=𝐶𝜔_𝑘,𝛾_𝑘,𝜈_𝑘. (3.19)