Helle HallikRational spline histopolation

Helle Hallik Rational spline histopolation

Tartu 2015 ISSN 1024-4212 ISBN 978-9949-32-834-5

DISSERTATIONES MATHEMATICAE UNIVERSITATIS TARTUENSIS

98

Helle Hallik

Rational spline histopolation

DISSERTATIONES MATHEMATICAE UNIVERSITATIS TARTUENSIS

98

DISSERTATIONES MATHEMATICAE UNIVERSITATIS TARTUENSIS 98

HELLE HALLIK

Rational spline histopolation

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1 + d _i (x − x _i−1 )

-

1 + d _i (x − x _i−1 ) > 0 x ∈ [x _i−1 , x _i ]

i = 1, . . . , n,

D

x i

x _i−1

S(x)dx = z _i (x _i − x _i−1 ) , i = 1, . . . , n .

! -

S (x ₀ ) = α , S (x _n ) = β

S(x ₀ ) = α , S(x _n ) = β

α

β

A

x ₀

x _n

%

[x _i−1 , x _i ]

S (x) = b _i − a _i d _i

(1 + d _i (x − x _i−1 )) ² ,

-

S

C ¹ [a, b]

[a, b]

.". -."5 ."4

z ₁ < . . . < z _n or z ₁ > . . . > z _n or z ₁ = . . . = z _n ,

-

z _i

S (x ₀ ) = α > 0

S (x _n ) = β > 0

S(x ₀ ) = α < z ₁

S(x _n ) = β > z _n

23 3

C ¹

+

S ₁

S ₂

[x _i−1 , x _i ]

S ₁ (x) = a _1i + b _1i (x − x _i−1 )

1 + d _1i (x − x _i−1 )

S ₂ (x) = a _2i + b _2i (x − x _i−1 ) 1 + d _2i (x − x _i−1 ) ,

x ∈ [x _i−1 , x _i ]

c _1i = b _1i − a _1i d _1i

c _2i = b _2i − a _2i d _2i

-

g (x) = c _1i

(1 + d _1i (x − x _i−1 )) ² − c _2i

(1 + d _2i (x − x _i−1 )) ² .

>

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c _1i

(1 + d _1i (x − x _i−1 )) ² = c _2i

(1 + d _2i (x − x _i−1 )) ² .

."1

1 + d _1i (x − x _i−1 ) > 0

1+d _2i (x − x _i−1 ) > 0

c _1i

c _2i

c _1i = c _2i = 0

g (x) = 0

[x _i−1 , x _i ]

c _1i c _2i > 0

g (x) = 0

1 + d _1i (x − x _i−1 ) 1 + d _2i (x − x _i−1 ) =

c _1i c _2i

_1/2

1 + d _1i (x − x _i−1 ) = c _1i

c _{2i 1/2}

(1 + d _2i (x − x _i−1 )).

+ > @

[x _i−1 , x _i ]

-" -

g (x) = 0

g

n

[x ₀ , x _n ]

S ₁

S ₂

z _i

"

x _i

x i−1

g(x)dx = 0 , i = 1, . . . , n ,

-

ξ _i ∈ (x _i−1 , x _i )

g(ξ _i ) = 0

[x _i−1 , x _i ]

g (x) = 0

."4 -

g(x ₀ ) = 0

g(x _n ) = 0

g

n + 2

[x ₀ , x _n ]

g

n + 1

(x ₀ , x _n )

-

g

n

."5 -

g (x ₀ ) = 0

g (x _n ) = 0

g

n − 1

ξ _i

i = 1, . . . , n

n + 1

g

.". - ."5 ."4 -

g

n + 1

[x ₀ , x _n ]

!

[x _i−1 , x _i ]

g (x) = 0

?> & -

g

G-" ! &

x _i

i = 1, . . . , n − 1

-

g (x _i ) = 0

g (x) = 0

[x ₀ , x _n ]

g

g(x) = 0

c _i = b _i − a _i d _i

i = 1, . . . , n

."1

[x _i−1 , x _i ]

S (x) = c _i

(1 + d _i (x − x _i−1 )) ² .

+

S (x _i ) = m _i

i = 0, . . . , n

h _i = x _i − x _i−1

i = 1, . . . , n

."8 -

c _i = m _i−1 and c _i

(1 + d _i h _i ) ² = m _i .

! .". -

S

m _i

m _i > 0

m _i < 0

m _i = 0

i = 0, . . . , n

!

m _i = 0

m _i−1

m _i = (1 + h _i d _i ) ² ,

-

d _i = 1 h _i

m _i−1 m _i

_1/2

− 1

.

+

d _i = 0

m _i−1 = m _i

S(x) = b _i

d _i − c _i

d _i (1 + d _i (x − x _i−1 )) .

_{x i}

x i−1

S(x)dx = _{x i}

x i−1

b _i

d _i − c _i

d _i (1 + d _i (x − x _i−1 ))

dx

= b _i d _i h _i − c _i

d ² _i log(1 + h _i d _i ),

.". -

b _i d _i − c _i

h _i d ² _i log(1 + h _i d _i ) = z _i .

>."1. -

b _i

d _i = z _i + c _i

h _i d ² _i log(1 + h _i d _i ).

."11

S(x) = z _i + m _i−1

h _i d ² _i log(1 + h _i d _i ) − m _i−1

d _i (1 + d _i (x − x _i−1 )) ,

) >."10

S(x) = z _i + h _i m _{i−1 m}

m i−1 i

_1/2

− 1 ₂ log

m _i−1 m _i

_1/2

− h _i m _i−1

m i−1

m i

_1/2

− 1 1 + t m i−1

m i

_1/2

− 1 ,

."14

x ∈ [x _i−1 , x _i ], i = 1, . . . , n, t = (x − x _i−1 )/h _i .

!

d _i = 0

m _i−1 = m _i

b _i = c _i = m _i

a _i = z _i − h _i m _i /2

x ∈ [x _i−1 , x _i ]

S(x) = z _i + m _i

x −

x _i−1 + h _i 2

.

m _i = 0

S(x) = z _i

x ∈ [a, b]

."12 "

S

m _i

S

S

x ₁ , . . . , x _n−1

>"

E

m _i > 0

i

m _i < 0

i

" +

d _i = 0

m _i−1 = m _i (1 + h _i d _i ) ²

."15

S(x _i − 0) = z _i + h _i m _i (1 + h _i d _i ) ² log(1 + h _i d _i ) − h _i d _i (1 + h _i d _i ) h ² _i d ² _i

= z _i + h _i m _i (1 + h _i d _i ) ² (log(1 + h _i d _i ) − 1) + 1 + h _i d _i h ² _i d ² _i

= z _i + h _i m _i m _i−1

m _i

log m _i−1

m _{i 1/2}

− 1

+ m _i−1

m _{i 1/2}

m _i−1 m _i

1/2

− 1

₂ .

$ ."15 -

S(x _i−1 + 0) = z _i + h _i m _i−1 log(1 + h _i d _i ) − h _i d _i h ² _i d ² _i

= z _i − h _i m _i−1 (1 + h _i d _i ) ⁻²

log(1 + h _i d _i ) ⁻¹ − 1

+ (1 + h _i d _i ) ⁻¹

((1 + h _i d _i ) ⁻¹ − 1) ²

= z _i − h _i m _i−1 m _i m _i−1

log

m _i m _i−1

_1/2

− 1

+ m _i

m _{i−1 1/2}

m _i m _i−1

_1/2

− 1

₂ .

."12 -

d _i = 0 S(x _i − 0) = z _i + h _i

2 m _i , S(x _i−1 + 0) = z _i − h _i 2 m _i .

+

ϕ(x) =

⎧ ⎪

⎨

⎪ ⎩

x ² (log x − 1) + x

(x − 1) ² for x > 0 , x = 1 , 1

2 for x = 1 .

/ 3 3 "

#

ϕ(x) > 0

ϕ (x) > 0

ϕ (x) < 0

x > 0

lim

x→1 ϕ(x) = 1 2

lim

x→1 ϕ (x) = 1 3

lim

x→0+

ϕ(x)

x = 1

lim

x→∞

ϕ(x) log x = 1

1

2 < ϕ(x)

x < 1

0 < x < 1

%

ϕ(x ^1/2 ) log x

x 1.84.

1 $

x > 0

(x ² (log x − 1)+x)/(x − 1) ² .

:

x→0 lim + (x ² (log x − 1) + x) = lim

x→0 +

log x − 1 1 x ²

= lim

x→0 +

1 x

− 2 x ³

= lim

x→0 +

− x ² 2 = 0.

lim _{x→0 +} ϕ(x) = 0

'?

x > 0

x = 1

ϕ (x) =

(2x(log x − 1) + x ² · 1

x + 1)(x − 1) ²

(x − 1) ⁴ − (x ² (log x − 1) + x) · 2(x − 1) (x − 1) ⁴

= x ² − 2x log x − 1 (x − 1) ³ .

!

x > 1

(x − 1) ³ > 0

f (x) = x ² − 2x log x − 1

f(1) = 0

'-

f (x) = 2x − 2 log x − 2 = 2(x − log x − 1)

f (1) = 0

f (x) = 2 (1 − 1/x) > 0 x > 1

f (x) > 0 x > 1

f (x) > 0 x > 1

ϕ (x) > 0 x > 1

x→1 lim + ϕ (x) = lim

x→1 +

x ² − 2x log x − 1 (x − 1) ³ = lim

x→1 +

2 x ²

6 = 1 3 > 0.

+-

0 < x < 1

ϕ (x)

(x − 1) ³ < 0

f (x) = x ² − 2x log x − 1

lim _{x→0 +} f (x) = − 1

lim _{x→0 +} f (x) = ∞

f (x) = 2 (1 − 1/x) < 0 0 < x < 1

""

f

(0, 1)

f (1) = 0

f (x) > 0

(0, 1)

f

(0, 1)

f (x) < 0 0 < x < 1

ϕ (x) > 0

0 < x < 1

lim _{x→1 −} ϕ(x) = 1/3

ϕ (x) > 0 x > 0

ϕ

x > 0

lim _{x→0 +} ϕ(x) = 0,

--

ϕ(x) > 0 x > 0

-

ϕ

. E

x→1 lim

x=1

ϕ(x) = lim

x→1 x=1

=

2x(log x − 1) + x ² · 1 x + 1 2(x − 1)

= lim

x→1 x=1

2x log x − x + 1 2(x − 1)

= lim

x→1 x=1

2 log x + 2x · 1 x − 1 2

= lim

x→1 x=1

2 log x + 1

2 = 1

2 .

x→1 lim ϕ(x) = 1 2 .

ϕ

1

5 E

x→0 lim +

ϕ(x) x = lim

x→0 +

x(log x − 1) + 1 (x − 1) ² = 1

lim _{x→0 +} x log x = 0

E )-

ϕ(x) > 0 x > 0

x→∞ lim ϕ(x) log x = lim

x→∞

x ² (log x − 1) (x − 1) ² log x + lim

x→∞

x (x − 1) ² log x .

G

x → ∞

x→∞ lim ϕ(x) log x = lim

x→∞

x ²

(x − 1) ² · lim

x→∞

log x − 1 log x

- >" E

x→∞ lim ϕ(x) log x = 1.

4

ψ(x) = ϕ(x)/x = (x(log x − 1) + 1)/(x − 1) ²

ψ (x) =

((log x − 1) + x · 1

x )(x − 1) ² − (x(log x − 1) + 1)2(x − 1) (x − 1) ⁴

= − (x + 1) log x + 2(x − 1) (x − 1) ³ .

+

ψ (x) x ∈ (0, 1)

E

χ(x) = − (x + 1) log x + 2(x − 1)

"

lim _{x→0 +} χ(x) = ∞

χ(1) = 0

χ (x) = − 1/x + 1/x ² = ( − x + 1)/x ² > 0 0 < x < 1

χ

0 < x < 1

χ (1) = 0

χ (x) < 0 0 < x < 1

χ

0 < x < 1

χ(1) = 0

χ(x) > 0 0 < x < 1

ψ (x) = χ(x)/(x − 1) ³ < 0 0 < x < 1

ϕ(x)/x

0 < x < 1

. -

lim _{x→0 +} ϕ(x)/x = 1

lim _x→1 ϕ(x)/x = 1/2

"

2 5 -

x

ϕ(x)/ log x 2

ϕ(x) 2 log x

x ^∗

ϕ(x) = 2 log x

ϕ(1) = 1/2

log 1 = 0

lim _{x→1 +} ϕ(x)/ log x = ∞

x ^∗

x x ^∗

ϕ(x)/ log x 2

x ^∗ < √

1.84

x √

1.84

x x ^∗

ϕ(x)/ log x 2

E+."1 "

ϕ

S(x _i − 0) = z _i + h _i m _i ϕ

m _i−1 m _i

_1/2

."18

S (x _i−1 + 0) = z _i − h _i m _i−1 ϕ

m _i m _i−1

_1/2

.

S

S(x _i − 0) = S(x _i + 0)

i = 1, . . . , n − 1

m _i

h _i ϕ

m _i−1 m _i

_1/2

+ h _i+1 ϕ

m _i+1 m _i

_1/2

= δ _i ,

-

δ _i = z _i+1 − z _i

m _i > 0

S

-

δ _i > 0

!

d _i = 0

d _i+1 = 0

S(x _i − 0) = z _i + h _i

2 m _i

S(x _i−1 + 0) = z _i − h _i 2 m _i

-

m _i (h _i + h _i+1 ) = 2δ _i .

!

d _i = 0

d _i+1 = 0

d _i = 0

d _i+1 = 0

.".1 -

m _i h _i

2 + h _i+1 ϕ

m _i+1 m _i

1/2

= δ _i .

'.".. .".5 .".0 "

d _i = 0

m _i−1 = m _i

ϕ(1) = 1/2

( ."5 @?

m ₀ = α

m _n = β

."4

d ₀

d _n

z ₁ − h ₁ m ₀ ϕ

m ₁ m ₀

_1/2

= α , z _n + h _n m _n ϕ

m _n−1 m _n

_1/2

= β.

.".4

' >.".4 .".0 -

i = 0

i = n

h ₀ = 0

z ₀ = α

h _n+1 = 0

z _n+1 = β

? =

z _i

α

β

2 33 (

z _i

α

β

C ¹

δ _i = 0

.".5 .".4 .".0

m _i = 0

S

."5 ."4 " (

S

δ _i > 0

i = 1, . . . , n − 1

α > 0

β > 0

α < z ₁

β > z _n

E - > .".0 D.".5 - - 3

m _i = ϕ _i (m)

m = (m ₀ , . . . , m _n )

- -

[c, M ]

ϕ _i : [c, M] ⁿ⁺¹ → [c, M]

ϕ _i

m _i = ϕ _i (m)

i = 0, . . . , n

>.".0

m _i = ϕ _i (m) = δ _i

h _i ϕ _m

m i−1 i

_1/2

+ h _i+1 ϕ _m

m i+1 i

_1/2 .

m _i−1

m _i

m _i+1 ∈ [c, M ]

0 < c < M

ϕ

-

ϕ _i (m) δ _i

(h _i + h _i+1 )ϕ c

M

_1/2 .

( + ."1 4

ϕ((c/M) ^1/2 ) (c/M) ^1/2 /2

-

ϕ _i (m) 2δ _i h _i + h _i+1

M c

_1/2

M

2δ _i

h _i + h _i+1 (M c) ^1/2 .

% H

c

M

)

M/c 1.84

ϕ((M/c) ^1/2 ) 2 log(M/c) ^1/2

ϕ _i (m) δ _i

2(h _i + h _i+1 ) log ^M _c

_1/2 c

δ _i

h _i + h _i+1 c log M

c .

E > .".. .".5

.".0 -

> ."./ .".7 " .".4

."5 - ) >

α, β ∈ [c, M]

c

M

-

h ₀ = 0

h _n+1 = 0

A = max

0in

2δ _i

h _i + h _i+1 , B = min

0in

δ _i h _i + h _i+1 .

<

A = (M c) ^1/2

c

M

B c log(M/c)

!"

- )

ξ _i

(x _i−1 , x _i )

i = 1, . . . , n

S(ξ _i ) = f (ξ _i ) , i = 1, . . . , n ,

f

[a, b]

0 3 3 + * ,

f f ∈ Lip 1

S

C ¹

, ,

x _i = a + ih

h = (b − a)/n

i = 0, . . . , n

*

O(h ³ )

#% 3 3

O(h ³ )

1 $, ,

, 2 $ ',$

S (x ₀ ) = f ₀ = f (x ₀ )

S _i = S(x _i )

i = 0, 1

h ² f ₀ (S ₁ − f (ξ ₁ )) − h(f (ξ ₁ ) − S ₀ )(S ₁ − S ₀ ) = 0

2 ,# ./0

=

S

x _i

x i−1

S(x)dx =

x _i

x i−1

f (x)dx , i = 1, . . . , n ,

f

ξ _i ∈ (x _i−1 , x _i )

x i

x i−1

S(x) − f (x))dx = (S(ξ _i ) − f (ξ _i ))(x _i − x _i−1

= 0

-

S(ξ _i ) = f(ξ _i )

S

S (x ₀ ) = f (x ₀ ) , S (x _n ) = f (x _n )

S(x ₀ ) = f (x ₀ ) , S(x _n ) = f(x _n ) .

( ."1- -

2 33 ( * ,

f f ∈ Lip 1

$

f ∈ C ³ [a, b]

S

C ¹

*

O(h ³ )

E- 5"

# $

."14 ."12 > )-

m _i

i = 1, . . . , n − 1

- .".4 .".0

."4

m ₀ = α

m _n = β

ψ ₀ (m) ≡ m ₀ − α = 0 ,

ψ _i (m) ≡ m _i

h _i ϕ

m _i−1 m _i

_1/2

+ h _i+1 ϕ

m _i+1 m _i

_1/2

− δ _i = 0 , i = 1, . . . , n − 1 , ψ _n (m) ≡ m _n − β = 0

."5.

-@ ->.".4 " E-

Ψ(m) = (ψ ₀ (m), . . . , ψ _m (m))

Ψ(m) = 0

% - @ ."5. '-K " $

'-K

Ψ (m ^k )m ^k+1 = Ψ (m ^k )m ^k − Ψ(m ^k )

- ?" ?

Ψ

i

∂ψ _i

∂m _i−1 = 1 2 h _i 1

u _i ϕ (u _i ) ,

∂ψ _i

∂m _i = h _i ϕ(u _i ) − 1

2 h _i u _i ϕ (u _i ) + h _i+1 ϕ(v _i ) − 1

2 h _i+1 v _i ϕ (v _i ) ,

∂ψ _i

∂m _i+1 = 1 2 h _i+1 1

v _i ϕ (v _i )

-

u _i = (m _i−1 /m _i ) ^1/2

v _i = (m _i+1 /m _i ) ^1/2

0 33 (

u _i > 0

v _i > 0

∂ψ _i /∂m _i > 0

χ(x) = ϕ(x) − xϕ (x)/2

χ (x) = (ϕ (x) − xϕ (x))/2

χ (x) > 0 x > 0

χ(0) = 0

lim χ(x) = 0

x → 0 ₊

"

!

∂ψ _i /∂m _i−1 > 0

∂ψ _i /∂m _i+1 > 0

.".4 ->G " A

i

∂ψ _i

∂m _i − ∂ψ _i

∂m _i−1 − ∂ψ _i

∂m _i+1

= h _i

ϕ(u _i ) − 1 2

u _i + 1

u _i

ϕ (u _i )

+ h _i+1

ϕ(v _i ) − 1 2

v _i + 1 v _i

ϕ (v _i )

.

+

δ(x) = ϕ(x) − 1 2

x + 1

x

ϕ (x) .

δ (x) = 1 2

1 + 1 x ²

ϕ (x) − 1 2

x + 1

x

ϕ (x)

+ ."1

δ (x) > 0 x > 0

δ(x) = 0

>

2x ⁴ log x − 3x ⁴ + 4x ³ − 2x ² + 2x log x + 1 = 0

x ^∗ ≈ 0.734

0 33 , ' 78 , ,

m _i−1 /m _i (x ^∗ ) ² ≈ 0.54

m _i+1 /m _i (x ^∗ ) ²

+ )

n → ∞

m _i−1

m _i

m _i+1

"

! - - )- -

."14 ."12 " ."..

-- -

d _i = 0

m _i−1 = m _i

E- -"" B.6C

'-K "

033 9

B(m ⁰ , R)

,$

Ψ (x) − Ψ (y) L x − y ∀ x, y ∈ B(m ⁰ , R) ,

Ψ (m ⁰ ) ₋₁ b ₀ ,

Ψ (m ⁰ ) ₋₁

Ψ(m ⁰ ) b ₁

b ₀ b ₁ L 1/2

R (1 − √

1 − 2b ₀ )b ₁ /b ₀

m ^∗ ∈ B(m ⁰ , R)

+ +G

Ψ

>""

m ^k+1 − m ^∗ = O( m ^k − m ^{∗ 2} )

! +G

Ψ

ϕ

& ."5"

m ⁰

m ^k

> ."52 @

b ₁

-

f ∈ C ³ [a, b]

z _i = 1 h _i

x i

x i−1

f (x)dx , i = 1, . . . , n .

)

m _i = 2(z _i+1 − z _i )

h _i + h _i+1 , i = 1, . . . , n − 1 .

K

m ⁰ _i = f (x _i ) + O(h)

h = max h _i

m ⁰ _i = f (x _i ) + O(h ² )

-

m ^∗ _i = f (x _i ) + O(h)

m ⁰ _i − m ^∗ _i = O(h)

h

'-K - "