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spline.rar （3个子文件）

spline

Cubic Spline Interpolation.pdf 63KB

Cubic Spline Interpolation1.pdf 199KB

spline_interpolation.cpp 3KB

Cubic Spline Interpolation

Sky McKinley and Megan Levine

Math 45: Linear Algebra

Abstract. An introduction into the theory and application of cubic splines with accompanying Matlab

m-file cspline.m

Introduction

Real world numerical data is usually difficult to analyze. Any function which would

effectively correlate the data would be difficult to obtain and highly unwieldy. To this end,

the idea of the cubic spline was developed. Using this process, a series of unique cubic

polynomials are fitted between each of the data points, with the stipulation that the curve

obtained be continuous and appear smooth. These cubic splines can then be used to

determine rates of change and cumulative change over an interval. In this brief introduction,

we will only discuss splines which interpolate equally spaced data points, although a more

robust form could encompass unequally spaced points.

Theory

The fundamental idea behind cubic spline interpolation is based on the engineer’s tool

used to draw smooth curves through a number of points. This spline consists of weights

attached to a flat surface at the points to be connected. A flexible strip is then bent across

each of these weights, resulting in a pleasingly smooth curve.

The mathematical spline is similar in principle. The points, in this case, are numerical

data. The weights are the coefficients on the cubic polynomials used to interpolate the data.

These coefficients ’bend’ the line so that it passes through each of the data points without

any erratic behavior or breaks in continuity.

Process

The essential idea is to fit a piecewise function of the form

SÝxÞ =

ÝxÞ if x

² x < x

ÝxÞ if x

² x < x

n?1

ÝxÞ if x

n?1

² x < x

(1)

where s

is a third degree polynomial defined by

ÝxÞ = a

Ýx ? x

+ b

Ýx ? x

+ c

Ýx ? x

Þ + d

(2)

for i = 1,2,...,n ? 1.

The first and second derivatives of these n ? 1 equations are fundamental to this process,

and they are

ÝxÞ = 3a

Ýx ? x

+ 2b

Ýx ? x

Þ + c

ÝxÞ = 6a

Ýx ? x

Þ + 2b

(3)

(4)

for i = 1,2,...,n ? 1.

The Four Properties of Cubic Splines

Our spline will need to conform to the following stipulations.

1. The piecewise function SÝxÞ will interpolate all data points.

2. SÝxÞ will be continuous on the interval ßx

3. S

ÝxÞ will be continuous on the interval ßx

4. S

ÝxÞ will be continuous on the interval ßx

Since the piecewiece function SÝxÞ will interpolate all of the data points, we can

conclude that

SÝx

Þ = y

(5)

for i = 1,2,...,n ? 1. Since x

5 ßx

i+1

à, SÝx

Þ = s

Ýx

Þ and we can use equation (2) to

produce

= s

Ýx

= a

Ýx

? x

+ b

Ýx

? x

+ c

Ýx

? x

Þ + d

= d

(6)

for each i = 1,2,...,n ? 1.

Since the curve SÝxÞ must be continuous across its entire interval, it can be concluded

that each sub-function must join at the data points, so

Ýx

Þ = s

i?1

Ýx

Þ (7)

for i = 2,3,...,n.

From equation (2),

Ýx

Þ = d

and

i?1

Ýx

Þ = a

i?1

Ýx

? x

i?1

+ b

i?1

Ýx

? x

i?1

+ c

i?1

Ýx

? x

i?1

Þ + d

i?1

(8)

= a

i?1

Ýx

? x

i?1

+ b

i?1

Ýx

? x

i?1

+ c

i?1

Ýx

? x

i?1

Þ + d

i?1

(9)

for i = 2,3,...,n ? 1. Letting h = x

? x

i?1

in equation (8), we have

= a

i?1

+ b

i?1

+ c

i?1

h + d

i?1

(10)

for i = 2,3,...,n ? 1.

Also, to make the curve smooth across the interval, the derivatives must be equal at the

data points; that is,

Ýx

Þ = s

i?1

Ýx

Þ (11)

However, by equation (3),

Ýx

Þ = c

and

i?1

Ýx

Þ = 3a

i?1

Ýx

? x

i?1

+ 2b

i?1

Ýx

? x

i?1

Þ + c

i?1

= 3a

i?1

Ýx

? x

i?1

+ 2b

i?1

Ýx

? x

i?1

Þ + c

i?1

. (12)

Again, letting h = x

? x

i?1

, we arrive at

= 3a

i?1

+ 2b

i?1

h + c

i?1

(13)

for i = 2,3,u,n ? 1.

From equation (4), s

ÝxÞ = 6a

Ýx ? x

Þ + 2b

, so

ÝxÞ = 6a

Ýx ? x

Þ + 2b

Ýx

Þ = 6a

Ýx

? x

Þ + 2b

Ýx

Þ = 2b

(14)

for i = 2,3,...,n ? 2.

Lastly, since s

ÝxÞ has to be continuous across the interval, s

Ýx

Þ = s

i+1

Ýx

Þ for

i = 1,2,3,`,n ? 1. This and equation (14) lead us to the equation

Ýx

i+1

Þ = 6a

Ýx

i+1

? x

Þ + 2b

i+1

Ýx

i+1

Þ = 6a

Ýx

i+1

? x

Þ + 2b

(15)

(16)

and, letting h = x

i+1

? x

and using the conclusion from equations (14) and (16),

i+1

Ýx

i+1

Þ = 6a

Ýx

i+1

? x

Þ + 2b

i+1

= 6a

h + 2b

(17)

(18)

These equations can be much simplified by substituting M

for s

Ýx

Þ and expressing the

above equations in terms of M

and y

. This makes the determination of the weights a

, and d

a much easier task. Each b

can be represented by

Ýx

Þ = 2b

= 2b

(19)

and d

has already been determined to be

= y

. (20)

Similarly, using equation a

can be re-written as

i+1

= 6a

h + 2b

h = 2b

i+1

? 2b

i+1

? 2b

2Ý

i+1

Þ ? 2Ý

i+1

? M

(21)

and c

can be re-written as

i+1

= a

+ b

+ c

h + d

h = ?a

? b

? d

+ d

i+1

? b

? d

+ d

i+1

? b

+ d

i+1

= Ý?a

? b

hÞ ?

? d

i+1

= ?Ý

i+1

? M

hÞ ?

? y

i+1

? y

? Ý

i+1

? M

h +

hÞ

i+1

? y

? Ý

i+1

? M

+ 3M

Þh

i+1

? y

? Ý

i+1

+ 2M

Þh.

(22)

We now have our equations for determining the weights for our n ? 1 equations

i+1

? M

i+1

? y

? Ý

i+1

+ 2M

Þh

= y

(23)

These systems can be handled more conveniently by putting them into matrix form as

follows

i+1

= 3a

+ 2b

h + c

3Ý

i+1

? M

Þh

+ 2Ý

Þh +

i+1

? y

? Ý

i+1

+ 2M

Þh =

i+2

? y

i+1

? Ý

i+2

+ 2M

i+1

Þh

3Ý

i+1

? M

Þh

+ 2Ý

Þh ? Ý

i+1

+ 2M

Þh + Ý

i+2

+ 2M

i+1

Þh = ?Ý

i+1

? y

Þ +

i+2

? y

i+1

hÝ

i+1

? 3M

? Ý

i+1

+ 2M

Þ + Ý

i+2

+ 2M

i+1

ÞÞ =

? 2y

i+1

+ y

i+2

ÝM

+ 4M

i+1

+ M

i+2

Þ =

? 2y

i+1

+ y

i+2

+ 4M

i+1

+ M

i+2

= 6Ý

? 2y

i+1

+ y

i+2

(24)

for i = 1,2,3,`,n ? 1

which leads to the matrix equation

1 4 1 0 ` 0 0 0 0

0 1 4 1 ` 0 0 0 0

0 0 1 4 ` 0 0 0 0

_ _ _ _ b _ _ _ _

0 0 0 0 ` 4 1 0 0

0 0 0 0 ` 1 4 1 0

0 0 0 0 ` 0 1 4 1

n?3

n?2

n?1

? 2y

+ y

? 2y

+ y

? 2y

+ y

n?4

? 2y

n?3

+ y

n?2

n?3

? 2y

n?2

+ y

n?1

n?2

? 2y

n?1

+ y

(25)

Note that this system has n ? 2 rows and n columns, and is therefore under-determined. In

order to generate a unique cubic spline, two other conditions must be imposed upon the

system. This leads us to our next section.

Three types of Splines

Natural splines

This first spline type includes the stipulation that the second derivative be equal to zero

at the endpoints.

= M

= 0 (26)

This results in the spline extending as a line outside the endpoints. The matrix for

determining the M

? M

values can be adapted accordingly.

评论收藏

内容反馈

zhuhm96

2011-11-16

程序能运行，结果正确。
zhouchaochuang

2012-04-13

程序能够运行，研究源代码中~~~
xjieliu999

2011-11-26

写的太乱了，只是一个C程序，算不上是C++的，没有说明，看的有点累。
xiaochangjuan

2013-10-31

thanks,能运行
winter741dj

2015-05-03

对我很有帮助，谢谢楼主的分享

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