Abstract

By applying the matrix-exponential operator technique to the radiative-transfer equation in discrete form, new analytical solutions are obtained for the transmission and reflection matrices in the limiting cases x ≪ 1 and x ≫ 1, where x is the optical depth of the layer. Orthogonality of the eigenvectors of the matrix exponential apparently yields new conditions for determining Chandrasekhar’s characteristic roots. The exact law of reflection for the discrete eigenfunctions is also obtained. Finally, when used in conjunction with the doubling method, the matrix exponential should result in reductions in both computation time and loss of precision.

© 1981 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
    [Crossref]
  3. W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
    [Crossref]
  4. G. C. Wick, “Über ebene diffusionis Probleme,” Z. Phys. 121, 702–718 (1943).
    [Crossref]
  5. H. C. van de Hulst, “A new look at multiple scattering,” Tech. Rep. (NASA Inst. Space Studies, New York, 1963).
  6. K.-N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: applications to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303–1326 (1973).
    [Crossref]
  7. I. P. Grant and G. E. Hunt, “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London, Ser. A 313, 183–197 (1969); “II. Stability and non-negativity,”  313, 199–216 (1969).
    [Crossref]
  8. R. Redheffer, “Transmission-line theory and scattering and transfer,” J. Math. Phys. 41, 1–41 (1962).
  9. S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
    [Crossref]
  10. J. E. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
    [Crossref]
  11. J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. I. The doubling method,” J. Atmos. Sci. 28, 120–125 (1971).
    [Crossref]
  12. G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
    [Crossref] [PubMed]
  13. G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: Scattering from maritime haze,” Appl. Opt. 12, 1071–1084 (1973).
    [Crossref] [PubMed]
  14. Y. Inoue and S. Horowitz, “Numerical solution of full-wave equation with mode coupling,” Radio Sci. 1, 957–970 (1966).
  15. K. Suchy, “Attenuation of waves in plasmas,” Radio Sci. 7, 871–884 (1972).
    [Crossref]
  16. H. B. Keller, “Approximate solution of transport problems. Part II. Convergence and applications of the discrete-ordinate method,” SIAM J. Appl. Math. 8, 43–73 (1960).
    [Crossref]
  17. R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960), p. 165.
  18. F. R. Gantmacher, Theory of Matrices (Chelsea, New York, 1956), Vol. 1, pp. 116 ff.
  19. R. Aronson, “A theorem concerning reflection and transmission from a nonabsorbing slab,” Nucl Sci. Eng. 44, 449 (1971).
  20. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Massachusetts, 1967).
  21. G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media—I. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
    [Crossref]
  22. R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
    [Crossref]
  23. V. N. Faddeeva, Computational Methods of Linear Algebra, translated by C. D. Benster (Dover, New York, 1959), pp. 226 ff.

1975 (1)

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[Crossref]

1974 (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

1973 (4)

K.-N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: applications to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303–1326 (1973).
[Crossref]

G. N. Plass, G. W. Kattawar, and F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
[Crossref] [PubMed]

G. W. Kattawar, G. N. Plass, and F. E. Catchings, “Matrix operator theory of radiative transfer. 2: Scattering from maritime haze,” Appl. Opt. 12, 1071–1084 (1973).
[Crossref] [PubMed]

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media—I. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[Crossref]

1972 (1)

K. Suchy, “Attenuation of waves in plasmas,” Radio Sci. 7, 871–884 (1972).
[Crossref]

1971 (2)

R. Aronson, “A theorem concerning reflection and transmission from a nonabsorbing slab,” Nucl Sci. Eng. 44, 449 (1971).

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. I. The doubling method,” J. Atmos. Sci. 28, 120–125 (1971).
[Crossref]

1969 (2)

J. E. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[Crossref]

I. P. Grant and G. E. Hunt, “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London, Ser. A 313, 183–197 (1969); “II. Stability and non-negativity,”  313, 199–216 (1969).
[Crossref]

1966 (2)

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Y. Inoue and S. Horowitz, “Numerical solution of full-wave equation with mode coupling,” Radio Sci. 1, 957–970 (1966).

1962 (1)

R. Redheffer, “Transmission-line theory and scattering and transfer,” J. Math. Phys. 41, 1–41 (1962).

1960 (2)

H. B. Keller, “Approximate solution of transport problems. Part II. Convergence and applications of the discrete-ordinate method,” SIAM J. Appl. Math. 8, 43–73 (1960).
[Crossref]

R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
[Crossref]

1943 (1)

G. C. Wick, “Über ebene diffusionis Probleme,” Z. Phys. 121, 702–718 (1943).
[Crossref]

Aronson, R.

R. Aronson, “A theorem concerning reflection and transmission from a nonabsorbing slab,” Nucl Sci. Eng. 44, 449 (1971).

Bellman, R.

R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
[Crossref]

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960), p. 165.

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Massachusetts, 1967).

Catchings, F. E.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Faddeeva, V. N.

V. N. Faddeeva, Computational Methods of Linear Algebra, translated by C. D. Benster (Dover, New York, 1959), pp. 226 ff.

Gantmacher, F. R.

F. R. Gantmacher, Theory of Matrices (Chelsea, New York, 1956), Vol. 1, pp. 116 ff.

Grant, I. P.

I. P. Grant and G. E. Hunt, “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London, Ser. A 313, 183–197 (1969); “II. Stability and non-negativity,”  313, 199–216 (1969).
[Crossref]

Hansen, J. E.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. I. The doubling method,” J. Atmos. Sci. 28, 120–125 (1971).
[Crossref]

J. E. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[Crossref]

Horowitz, S.

Y. Inoue and S. Horowitz, “Numerical solution of full-wave equation with mode coupling,” Radio Sci. 1, 957–970 (1966).

Howell, H. B.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Hunt, G. E.

I. P. Grant and G. E. Hunt, “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London, Ser. A 313, 183–197 (1969); “II. Stability and non-negativity,”  313, 199–216 (1969).
[Crossref]

Inoue, Y.

Y. Inoue and S. Horowitz, “Numerical solution of full-wave equation with mode coupling,” Radio Sci. 1, 957–970 (1966).

Irvine, W. M.

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[Crossref]

Jacobowitz, H.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

Kalaba, R.

R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
[Crossref]

Kattawar, G. W.

Keller, H. B.

H. B. Keller, “Approximate solution of transport problems. Part II. Convergence and applications of the discrete-ordinate method,” SIAM J. Appl. Math. 8, 43–73 (1960).
[Crossref]

Liou, K.-N.

K.-N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: applications to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303–1326 (1973).
[Crossref]

Plass, G. N.

Redheffer, R.

R. Redheffer, “Transmission-line theory and scattering and transfer,” J. Math. Phys. 41, 1–41 (1962).

Suchy, K.

K. Suchy, “Attenuation of waves in plasmas,” Radio Sci. 7, 871–884 (1972).
[Crossref]

Travis, L. D.

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Twomey, S.

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, “A new look at multiple scattering,” Tech. Rep. (NASA Inst. Space Studies, New York, 1963).

Wick, G. C.

G. C. Wick, “Über ebene diffusionis Probleme,” Z. Phys. 121, 702–718 (1943).
[Crossref]

Wing, G. M.

R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
[Crossref]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Massachusetts, 1967).

Appl. Opt. (2)

Astrophys. J. (1)

J. E. Hansen, “Radiative transfer by doubling very thin layers,” Astrophys. J. 155, 565–573 (1969).
[Crossref]

Icarus (1)

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[Crossref]

J. Atmos. Sci. (3)

K.-N. Liou, “A numerical experiment on Chandrasekhar’s discrete-ordinate method for radiative transfer: applications to cloudy and hazy atmospheres,” J. Atmos. Sci. 30, 1303–1326 (1973).
[Crossref]

S. Twomey, H. Jacobowitz, and H. B. Howell, “Matrix methods for multiple-scattering problems,” J. Atmos. Sci. 23, 289–296 (1966).
[Crossref]

J. E. Hansen, “Multiple scattering of polarized light in planetary atmospheres. I. The doubling method,” J. Atmos. Sci. 28, 120–125 (1971).
[Crossref]

J. Math. Phys. (2)

R. Redheffer, “Transmission-line theory and scattering and transfer,” J. Math. Phys. 41, 1–41 (1962).

R. Bellman, R. Kalaba, and G. M. Wing, “Invariant imbedding and mathematical physics. I. Particle processes,” J. Math. Phys. 1, 280–308 (1960).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

G. W. Kattawar and G. N. Plass, “Interior radiances in optically deep absorbing media—I. Exact solutions for one-dimensional model,” J. Quant. Spectrosc. Radiat. Transfer 13, 1065–1080 (1973).
[Crossref]

Nucl Sci. Eng. (1)

R. Aronson, “A theorem concerning reflection and transmission from a nonabsorbing slab,” Nucl Sci. Eng. 44, 449 (1971).

Proc. R. Soc. London, Ser. A (1)

I. P. Grant and G. E. Hunt, “Discrete space theory of radiative transfer I. Fundamentals,” Proc. R. Soc. London, Ser. A 313, 183–197 (1969); “II. Stability and non-negativity,”  313, 199–216 (1969).
[Crossref]

Radio Sci. (2)

Y. Inoue and S. Horowitz, “Numerical solution of full-wave equation with mode coupling,” Radio Sci. 1, 957–970 (1966).

K. Suchy, “Attenuation of waves in plasmas,” Radio Sci. 7, 871–884 (1972).
[Crossref]

SIAM J. Appl. Math. (1)

H. B. Keller, “Approximate solution of transport problems. Part II. Convergence and applications of the discrete-ordinate method,” SIAM J. Appl. Math. 8, 43–73 (1960).
[Crossref]

Space Sci. Rev. (1)

J. E. Hansen and L. D. Travis, “Light scattering in planetary atmospheres,” Space Sci. Rev. 16, 527–610 (1974).
[Crossref]

Z. Phys. (1)

G. C. Wick, “Über ebene diffusionis Probleme,” Z. Phys. 121, 702–718 (1943).
[Crossref]

Other (6)

H. C. van de Hulst, “A new look at multiple scattering,” Tech. Rep. (NASA Inst. Space Studies, New York, 1963).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

R. Bellman, Introduction to Matrix Analysis (McGraw-Hill, New York, 1960), p. 165.

F. R. Gantmacher, Theory of Matrices (Chelsea, New York, 1956), Vol. 1, pp. 116 ff.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Massachusetts, 1967).

V. N. Faddeeva, Computational Methods of Linear Algebra, translated by C. D. Benster (Dover, New York, 1959), pp. 226 ff.

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Figures (4)

Fig. 1
Fig. 1

Geometry is shown for scattering–absorbing particles occupying the half-space x ≥ 0.

Fig. 2
Fig. 2

In the discrete problem, u and v characterize the right-going and left-going radiation, respectively.

Fig. 3
Fig. 3

The (1,1) matrix elements of T and R are shown as a function of optical depth x in the nonconservative case: points are exact, and the solid and dashed curves give the thick-layer and thin-layer approximations, respectively.

Fig. 4
Fig. 4

The thick-layer (solid curves) and thin-layer (dashed curves) approximations to T11 and R11 are plotted, along with exact values (points) versus optical depth x (conservative case).

Equations (134)

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μ Ψ ( x , μ ) / x = - Ψ ( x , μ ) + ( ω 0 / 2 ) - 1 + 1 d μ p ( μ , μ ) Ψ ( x , μ ) ,
μ i d Ψ ( x , μ i ) / d x = - Ψ ( x , μ i ) + ( ω 0 / 2 ) j p ( μ i , μ j ) a j Ψ ( x , μ j ) , i = 1 , 2 , , n , - 1 , - 2 , , - n ,
j = 1 n a j = 1.
a · b = a b = j a j b j .
( a b ) · ( c d ) = ( a b ) ( c d ) = a · c + b · d .
u i ( x ) = ( a i μ i ) 1 / 2 ψ ( x , μ i ) , v i ( x ) = ( a i μ i ) 1 / 2 ψ ( x , - μ i ) ,             i = 1 , 2 , , n .
M i j = δ i j μ j ,
A i j = δ i j a j ,             i , j = 1 , 2 , , n .
P = ( P 1 P 2 P 2 P 1 ) ,
d / d x ( u v ) = - ( Q 1 - Q 2 Q 2 - Q 1 ) ( u v ) = - Q ( u v ) ,
Q 1 M - 1 - ( ω 0 / 2 ) A 1 / 2 M - 1 / 2 P 1 M - 1 / 2 A 1 / 2 , Q 2 ( ω 0 / 2 ) A 1 / 2 M - 1 / 2 P 2 M - 1 / 2 A 1 / 2 ,
[ u ( x ) v ( x ) ] = e - Q x [ u ( 0 ) v ( 0 ) ] .
[ u ( x ) v ( 0 ) ] = ( T R R T ) [ u ( 0 ) v ( x ) ] .
[ u ( x ) v ( x ) ] = ( T - R T - 1 R R T - 1 - T - 1 R T - 1 ) [ u ( 0 ) v ( 0 ) ] .
e - Q x = ( T - R T - 1 R R T - 1 - T - 1 R T - 1 ) .
[ u ( x ) v ( x ) ] = e - Q x ( 0 1 1 0 ) [ v ( 0 ) u ( 0 ) ]
[ e - Q x ( 0 1 1 0 ) ] 2 = 1.
e - 2 Q x = ( e - Q x ) 2
( T 2 - R 2 T 2 - 1 R 2 R 2 T 2 - 1 - T 2 - 1 R 2 T 2 - 1 ) = ( T 1 - R 1 T 1 - 1 R 1 R 1 T 1 - 1 - T 1 - 1 R 1 T 1 - 1 ) 2 .
T 2 = T 1 ( 1 - R 1 2 ) - 1 T 1 , R 2 = R 1 + T 1 R 1 ( 1 - R 1 2 ) - 1 T 1 .
e - Q x 1 - Q x + ( 1 / 2 ) Q 2 x 2 - ( 1 / 6 ) Q 3 x 3 .
T - 1 1 + Q 1 x + ( 1 / 2 ) ( Q 1 2 - Q 2 2 ) x 2 - ( 1 / 6 ) [ Q 2 ( Q 2 Q 1 - Q 1 Q 2 ) - Q 1 ( Q 1 2 - Q 2 2 ) ] x 3 ,
T 1 - Q 1 x + ( 1 / 2 ) ( Q 1 2 + Q 2 2 ) x 2 - ( 1 / 6 ) [ Q 1 3 + Q 2 Q 1 Q 2 + 2 ( Q 1 Q 2 2 + Q 2 2 Q 1 ) ] x 3 .
R Q 2 x - ( 1 / 2 ) ( Q 1 Q 2 + Q 2 Q 1 ) x 2 + ( 1 / 6 ) ( Q 1 2 Q 2 + Q 2 Q 1 2 + 2 Q 2 3 + 2 Q 1 Q 2 Q 1 ) x 3 .
T direct - M - 1 x + ( 1 / 2 ) ( M - 1 x ) 2 - ( 1 / 6 ) ( M - 1 x ) 3 ,
T direct = e - M - 1 x .
( T direct ) i j = δ i j e - x / μ j ,
( c 0 ) i = ( a i / μ i ) 1 / 2 , ( c 1 ) i = ( a i μ i ) 1 / 2 , ( c 2 ) i = ( a i μ i 3 ) 1 / 2 ,             i = 1 , 2 , , n .
J = ( 1 / 2 ) - 1 + 1 ψ ( x , μ ) d μ = ( 1 / 2 ) ( u + v ) · c 0 ,
F = 2 - 1 + 1 ψ ( x , μ ) μ d μ = 2 ( u - v ) · c 1 .
d F / d x = - 4 ( 1 - ω 0 ) J .
( d / d x ) ( u - v ) · c 1 = 0 ( d / d x ) ( u + v ) · c 2 = 0 ,
T ¯ 0 1 ψ ( x , μ ) μ d μ = c 1 · u ( x ) , R ¯ - 1 0 ψ ( 0 , μ ) μ d μ = c 1 · v ( 0 ) .
c 1 · u ( 0 ) .
T ¯ + R ¯ = c 1 · ( T + R ) u ( 0 ) ,
F abs = c 1 · ( 1 - T - R ) u ( 0 ) .
c 1 · ( 1 - T - R ) u ( 0 ) = u ( 0 ) · ( 1 - T - R ) c 1 = 0.
( T + R ) c 1 = c 1 .
ψ ( x , μ i ) + ψ ( 0 , - μ i ) = 1 ,             i = 1 , , n ,
( u v ) = e - k j x ( f j g j ) and e k j x ( g j f j ) ,             j = 1 , , n .
( f j ) i = ( a i μ i ) ( 1 / 2 ) / ( 1 - k j μ i ) ,
( g j ) i = ( a i μ i ) ( 1 / 2 ) / ( 1 + k j μ i ) ,             i = 1 , , n ,
ω 0 i = 1 n a i / ( 1 - k 2 μ i 2 ) = 1 ,
( f j g j )
( a b )
e - k j x ( f j g j ) ( a b ) = e - k j x ( f j a f j b g i a g j b )
( f j - g j )
C j 2 ( f j g j ) · ( f j - g j ) = f j 2 - g j 2 = 4 k j s = 1 n a s [ μ s / ( 1 - k j 2 μ s 2 ) ] 2 ( isotropic case ) .
( f ˆ i ĝ i ) · ( f ˆ j - ĝ j ) = δ i j , ( f ˆ i ĝ i ) · ( - ĝ j f ˆ j ) = 0 ,
( f ˆ i - ĝ i ) · ( f ˆ j + ĝ j ) = δ i j , f ˆ i · ĝ j = f ˆ j · ĝ i .
s = 1 n a s μ s 2 / ( 1 - k i 2 μ s 2 ) ( 1 - k j 2 μ s 2 ) = 0 ,             i j .
e - Q x = j = 1 n [ e k j x ( ĝ j f ˆ j ) ( - g j f j ) + e - k j x ( f ˆ j ĝ j ) ( f ˆ j - ĝ j ) ] ,
i ( f ˆ i - ĝ i ) ( f ˆ i + ĝ i ) = 1 , i ( f ˆ i ĝ i - ĝ i f ˆ i ) = 0 ,
T - 1 = j = 1 n ( e k j x f ˆ j f ˆ j - e - k j x ĝ j ĝ j ) .
T - 1 ~ e k 1 x f ˆ 1 f ˆ 1 .
R T - 1 = j = 1 n ( e k j x ĝ j f ˆ j - e - k j x f ˆ j ĝ j ) .
ĥ i = m ( F - 1 ) i m f ˆ m ,
ĥ i · f ˆ j = m ( F - 1 ) i m f ˆ m · f ˆ j = ( F - 1 F ) i j = δ i j .
h j = W f j ,             j = 1 , 2 , , n .
( 1 - k j μ s ) ( f j ) s = ( a s μ s ) 1 / 2 .
s ( 1 / μ s ) w s ( f i ) s ( 1 - k j μ s ) ( f j ) s = s ( a s / μ s ) 1 / 2 w s ( f i ) s .
( k i - k j ) s w s ( f i ) s ( f j ) s = ( k i - k j ) ( h i · f j ) = s ( a s / μ s ) 1 / 2 ( f i - f j ) s w s ,
s ( a s / μ s ) 1 / 2 ( f i ) s w s
s ( a s / μ s ) 1 / 2 ( f i ) s w s = 1 ,             i = 1 , 2 , , n ,
T ( 0 ) = j e - k j x ĥ j ĥ j .
T - 1 T ( 0 ) = j f ˆ j ĥ j - i , j e - ( k i + k j ) x ĝ j ĝ j ĥ i ĥ i = 1 - i , j e - ( k i + k j ) x ( ĥ i · ĝ j ) ĝ j ĥ i ,
T ( 1 ) = T ( 0 ) [ 1 + i , j e - ( k i + k j ) x ( ĥ i · ĝ j ) ĝ j ĥ i ] = j e - k j x ĥ j ĥ j + i , j , m e - ( k i + k j + k m ) x × ( ĥ i · ĝ j ) ( ĥ m · ĝ j ) ĥ m ĥ i .
R ( 0 ) = j ĝ j ĥ j = i , j ( f ˆ i · ĝ j ) ĥ i ĥ j ,
R ( 1 ) = j ĝ j ĥ j - i , j e - ( k i + k j ) x ( ĝ i · ĥ j ) f ˆ i ĥ j + i , j , m e - ( k j + k m ) x ( ĝ j · ĥ i ) ( ĝ j · ĥ m ) ĝ i ĥ m .
j ( ĥ i · ĝ j ) ĝ j = f ˆ i - ĥ i , ĝ i · ĥ j = ĝ j · ĥ i ,
R ( 1 ) = i , j [ ( f ˆ i · ĝ j ) - ( ĝ i · ĥ j ) e - ( k i + k j ) x ] ĥ i ĥ j .
ĝ i = j ( ĝ i · ĥ j ) f ˆ j
ĝ i - j ( ĥ i · ĝ j ) f ˆ j = 0 ,
( f ˆ i ĝ i ) - j ( ĥ i · ĝ j ) ( ĝ j f ˆ j ) = ( ĥ i 0 ) .
[ u ( x ) v ( x ) ] i = { ( f ˆ i ĝ i ) e - k i x - j ( ĥ i · ĝ j ) ( ĝ j f ˆ j ) e k j x ,             x 0 ( ĥ i 0 ) ,             x = 0
e - k j x ĥ j ĥ j .
ĝ j ĥ j
k 1 = [ ( 15 + 97 ) / 8 ] 1 / 2 = 1.7624 , k 2 = [ ( 15 - 97 ) / 8 ] 1 / 2 = 0.8024 ,
f ˆ 1 = ( - 0.2160 0.9802 ) , f ˆ 2 = ( 0.9841 0.2296 ) , ĝ 1 = ( 0.0596 0.0619 ) , ĝ 2 = ( 0.1079 0.0981 ) .
ĥ 1 = ( - 0.2264 0.9703 ) , ĥ 2 = ( 0.9664 0.2129 ) .
T ( 1 ) = ( 1.0022 - 0.0004 - 0.0004 0.9996 ) .
R ( 1 ) = ( 0.0025 0.0023 0.0023 0.0022 ) .
P 1 = P 2 = ( 1 1 1 1 ) ,
Q 1 = 1 8 ( 7 - 2 - 2 14 ) , Q 2 = 1 8 ( 1 2 2 2 ) .
e - Q x = M 0 - M 1 x + g = 1 n - 1 [ e k j x ( ĝ j f ˆ j ) ( - ĝ j f ˆ j ) + e - k j x ( f ˆ j ĝ j ) ( f ˆ j - ĝ j ) ] ,
( f ˆ j ĝ j ) · ( c 1 - c 1 ) = ( f ˆ j ĝ j ) · ( c 2 c 2 ) = 0 ,             j = 1 , , n - 1.
( c 1 c 1 ) · ( c 1 - c 1 ) = ( c 2 - c 2 ) · ( c 2 c 2 ) = 0.
M 0 2 = M 0 , M 0 M 1 + M 1 M 0 = 2 M 1 , M 1 2 = 0 ,
( c 1 c 1 ) ( c 2 c 2 ) + ( c 2 - c 2 ) ( c 1 - c 1 ) .
M 0 = ( ĉ 1 ĉ 1 ) ( ĉ 2 ĉ 2 ) + ( ĉ 2 - ĉ 2 ) ( ĉ 1 - ĉ 1 ) ,
M 1 = ( ĉ 1 ĉ 1 ) ( ĉ 1 - ĉ 1 ) .
1 = ( ĉ 1 ĉ 1 ) ( ĉ 2 ĉ 2 ) + ( ĉ 2 - ĉ 2 ) ( ĉ 1 - ĉ 1 ) + j [ ( ĝ j f ˆ j ) ( - ĝ j f ˆ j ) + ( f ˆ j ĝ j ) ( f ˆ j - ĝ j ) ] ,
Q = ( ĉ 1 ĉ 1 ) ( ĉ 1 - ĉ 1 ) + j [ - k j ( ĝ j f ˆ j ) ( - ĝ j f ˆ j ) + k j ( f ˆ j ĝ j ) ( f ˆ j - ĝ j ) ] .
T - 1 = ( ĉ 1 ĉ 2 + ĉ 2 ĉ 1 ) + x ĉ 1 ĉ 1 + j ( e k j x f ˆ j f ˆ j - e - k j x ĝ j ĝ j ) .
ĥ i · f ˆ j = δ i j ,             for j = 1 , , n - 1 , ĥ i · ĉ 1 = δ i n ,
T ( 0 ) = ( 1 / x ) ĥ n ĥ n + j e - k j x ĥ j ĥ j .
T - 1 T ( 0 ) 1 + ( 1 / x ) [ ( ĉ 2 · ĥ n ) ĉ 1 + ĉ 2 ] ĥ n .
T ( 1 ) = T ( 0 ) { 1 + ( 1 / x ) [ ( ĉ 2 · ĥ n ) ĉ 1 + ĉ 2 ] ĥ n } - 1 ( 1 / x ) ĥ n ĥ n { 1 - ( 1 / x ) [ ( ĉ 2 · ĥ n ) ĉ 1 + ĉ 2 ] ĥ n + 2 ( ĉ 2 · ĥ n ) ( 1 / x ) 2 [ ( ĉ 2 · ĥ n ) ĉ 1 + ĉ 2 ] ĥ n - } = ( 1 / x ) ĥ n { 1 - ( 1 / x ) 2 ( ĉ 2 · ĥ n ) + [ ( 1 / x ) 2 ( ĉ 2 · ĥ n ) ] 2 - } ĥ n .
T ( 1 ) = [ x + 2 ( ĉ 2 · ĥ n ) ] - 1 ĥ n ĥ n .
R T - 1 = ( ĉ 1 ĉ 2 - ĉ 2 ĉ 1 ) + x ĉ 1 ĉ 1 + j ( e k j x ĝ j f ˆ j - e - k j x f ˆ j ĝ j ) ,
R ( 0 ) = ( R T - 1 ) T ( 0 ) = j ĝ j ĥ j + ĉ 1 ĥ n ,
1 = j = 1 n - 1 f ˆ j ĥ j + ĉ 1 ĥ n ,
1 - R ( 0 ) = j = 1 n - 1 ( f ˆ j - ĝ j ) ĥ j .
c 1 = 1 2 lim ω 0 1 ( f n + g n ) ,
ĉ 1 · [ 1 - ( T + R ) ] u ( 0 ) = ĉ 1 · [ 1 - R ( 0 ) ] u ( 0 ) = ĉ 1 · g = 1 n - 1 ( f ˆ j - ĝ j ) ĥ j u ( 0 ) ,
R ( 1 ) = j ĝ j ĥ j + ĉ 1 ĥ n - [ x + 2 ( ĉ 2 · ĥ n ) ] - 1 ĥ n ĥ n .
f ˆ 1 = ( - 0.4589 0.9007 ) , ĝ 1 = ( 0.1033 0.1053 ) ,
ĉ 1 = ( 0.6324 0.4472 ) , ĉ 2 = ( 0.6324 0.2236 ) ,
ĥ 1 = ( - 0.5771 0.8161 ) , ĥ 2 = ( 1.162 0.5922 ) .
T ( 1 ) = ( x + 1.734 ) - 1 ( 1.350 0.6881 0.6881 0.3507 ) ,
R ( 1 ) = ( 0.6752 0.4588 0.4588 0.3507 ) - T ( 1 ) .
Q 1 = 1 4 ( 3 - 2 - 2 6 ) , Q 2 = 1 4 ( 1 2 2 2 ) .
T 2 T 1 2 .
exp ( - Q x ) = exp [ - ( Q 1 - Q 2 Q 2 - Q 1 ) x ] .
S = ( 1 / 2 ) ( 1 1 1 - 1 )
S = S , S S = 1 , S - 1 = S .
Q ¯ = S Q S - 1 = ( 0 Q 1 + Q 2 Q 1 - Q 2 0 ) ,
Q ¯ 2 = ( U 0 0 U ) ,
U ( Q 1 + Q 2 ) ( Q 1 - Q 2 ) .
S exp ( - Q x ) S - 1 = exp ( - Q ¯ x ) = cosh Q ¯ x - Q ¯ ( Q ¯ - 1 sinh Q ¯ x ) ,
cosh Q ¯ x = s = 0 ( Q ¯ 2 ) s x 2 s / ( 2 s ) ! = s = 0 ( U 0 0 U ) s x 2 s / ( 2 s ) ! ,
Q ¯ - 1 sinh Q ¯ x = s = 0 ( Q ¯ 2 ) s x 2 s + 1 / ( 2 s + 1 ) ! = s = 0 ( U 0 0 U ) s x 2 s + 1 / ( 2 s + 1 ) !
exp ( - Q x ) = S - 1 exp ( - Q ¯ x ) S .
d R / d x = Q 2 - ( Q 1 R + R Q 1 ) + R Q 2 R , d T / d x = T ( - Q 1 + Q 2 R ) ,
A u i = λ i u i , A v i = λ i v i ,             i = 1 , 2 , , n .
v j A u i = λ i ( u i · v j ) = λ j ( u i · v j ) ,
u i · v j = 0 ( λ i λ j ) .
u ˆ i · v ˆ j = δ i j .
A ( u ˆ 1 u ˆ 2 u ˆ n ) = ( u ˆ 1 u ˆ 2 u ˆ n ) ( λ 1 0 0 λ 2 λ n ) .
( v ˆ 1 v ˆ 2 v ˆ n )
A = ( u ˆ 1 u ˆ 2 u ˆ n ) ( λ 1 0 0 λ 2 λ n ) ( v ˆ 1 v ˆ 2 v ˆ n ) .
A = i λ i u ˆ i v ˆ i
f ( A ) = i f ( λ i ) u ˆ i v ˆ i .
1 = i u ˆ i v ˆ i .