Abstract

A generalized numerical wave-front reconstruction method is proposed that is suitable for diversified irregular pupil shapes of optical systems to be measured. That is, to make a generalized and regular normal equation set, the test domain is extended to a regular square shape. The compatibility of this method is discussed in detail, and efficient algorithms (such as the Cholesky method) for solving this normal equation set are given. In addition, the authors give strict analyses of not only the error propagation in the wave-front estimate but also of the discretization errors of this domain extension algorithm. Finally, some application examples are given to demonstrate this algorithm.

© 2000 Optical Society of America

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References

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  1. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  2. K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861 (1986).
    [CrossRef]
  3. J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
    [CrossRef]
  4. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  5. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  6. R. J. Noll, “Phase estimates from slope-type wavefront sensors,” J. Opt. Soc. Am. 68, 139–140 (1978).
    [CrossRef]
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase difference,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. J. Hermann, “Least-squares wave front errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  9. F. Roddier, C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991).
    [CrossRef] [PubMed]
  10. R. G. Lane, M. Tallon, “Wave-front reconstruction using a Shack–Hartmann sensor,” Appl. Opt. 31, 6902–6908 (1992).
    [CrossRef] [PubMed]
  11. I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).
  12. E. T. Pearson, “Hartmann test data reduction,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. SPIE1236, 628–632 (1990).
    [CrossRef]
  13. K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).
  14. D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.
  15. D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
    [CrossRef]
  16. C. Chen, Equations of Mathematical Physics (China High Educational Press, Beijing, 1992).
  17. J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1994) (Chinese edition).
  18. R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997).
    [CrossRef]
  19. W. Zou, “Figure control of large segmented mirror telescope,” Master of Science thesis (Nanjing Astronomical Instruments Research Center, Chinese Academy of Sciences, Nanjing, China, 1996), p. 7.
  20. B. Jian, “New method for shearing wavefront reconstruction and its applications in wave aberration evaluation,” Ph.D. dissertation (Zhejiang University, Hangzhou, China, 1995), p. 6.
  21. M. Zhang, Department of Applied Mathematics, Southeast University, Nanjing 210096, China (personal communication, 1999).

1992 (1)

1991 (1)

1986 (1)

K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861 (1986).
[CrossRef]

1980 (2)

1979 (1)

1978 (1)

1977 (3)

Bulirsch, R.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1994) (Chinese edition).

Chen, C.

C. Chen, Equations of Mathematical Physics (China High Educational Press, Beijing, 1992).

Freischlad, K.

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

Freischlad, K. R.

K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861 (1986).
[CrossRef]

Fried, D. L.

Ghozeil, I.

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

Hardy, J. W.

Hermann, J.

Hudgin, R. H.

Hunt, B. R.

Jian, B.

B. Jian, “New method for shearing wavefront reconstruction and its applications in wave aberration evaluation,” Ph.D. dissertation (Zhejiang University, Hangzhou, China, 1995), p. 6.

Jiang, S.

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

Koliopoulos, C. L.

K. R. Freischlad, C. L. Koliopoulos, “Modal estimation of a wave front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861 (1986).
[CrossRef]

J. W. Hardy, J. E. Lefebvre, C. L. Koliopoulos, “Real-time atmospheric compensation,” J. Opt. Soc. Am. 67, 360–369 (1977).
[CrossRef]

Lane, R. G.

Lefebvre, J. E.

Levine, B. M.

R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997).
[CrossRef]

McNeil, B.

R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997).
[CrossRef]

Noll, R. J.

Pearson, E. T.

E. T. Pearson, “Hartmann test data reduction,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. SPIE1236, 628–632 (1990).
[CrossRef]

Roddier, C.

Roddier, F.

Shao, L.

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.

Southwell, W. H.

Stoer, J.

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1994) (Chinese edition).

Su, D.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.

Tallon, M.

Yang, S.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

Zhang, H.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

Zhang, M.

M. Zhang, Department of Applied Mathematics, Southeast University, Nanjing 210096, China (personal communication, 1999).

Zhu, Q.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

Zielinski, R. J.

R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997).
[CrossRef]

Zou, W.

W. Zou, “Figure control of large segmented mirror telescope,” Master of Science thesis (Nanjing Astronomical Instruments Research Center, Chinese Academy of Sciences, Nanjing, China, 1996), p. 7.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (8)

Other (11)

I. Ghozeil, “Hartmann and other screen tests,” in Optical Shop Testing, D. Malacara, ed. (Wiley, New York, 1978).

E. T. Pearson, “Hartmann test data reduction,” in Advanced Technology Optical Telescopes IV, L. D. Barr, ed., Proc. SPIE1236, 628–632 (1990).
[CrossRef]

K. Freischlad, “Wavefront integration from difference data,” in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, G. T. Reid, eds., Proc. SPIE1755, 212–218 (1992).

D. Su, S. Jiang, L. Shao, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Proceedings of the European Southern Observatory Conference on Progress in Telescope and Instrumentation Technologies, M. H. Ulrich, ed. (European Southern Observatory, Garching, Germany, 1992), pp. 289–292.

D. Su, S. Jiang, W. Zou, S. Yang, S. Yang, H. Zhang, Q. Zhu, “Experiment system of thin-mirror active optics,” in Advanced Technology Optical Telescopes V, L. M. Stepp, ed., Proc. SPIE2199, 609–621 (1994).
[CrossRef]

C. Chen, Equations of Mathematical Physics (China High Educational Press, Beijing, 1992).

J. Stoer, R. Bulirsch, Introduction to Numerical Analysis (Springer-Verlag, New York, 1994) (Chinese edition).

R. J. Zielinski, B. M. Levine, B. McNeil, “Hartmann sensors for optical testing,” in Optical Manufacturing and Testing II, H. Stahl, ed., Proc. SPIE3134, 398–406 (1997).
[CrossRef]

W. Zou, “Figure control of large segmented mirror telescope,” Master of Science thesis (Nanjing Astronomical Instruments Research Center, Chinese Academy of Sciences, Nanjing, China, 1996), p. 7.

B. Jian, “New method for shearing wavefront reconstruction and its applications in wave aberration evaluation,” Ph.D. dissertation (Zhejiang University, Hangzhou, China, 1995), p. 6.

M. Zhang, Department of Applied Mathematics, Southeast University, Nanjing 210096, China (personal communication, 1999).

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Figures (14)

Fig. 1
Fig. 1

Schematic configuration of the optical system of the S–H test. 1, Pinhole reference source; 2, focus of the optical system to be tested; 3, beam-splitter cube; 4, collimator; 5, lenslet array; 6, reducing system; 7, CCD target.

Fig. 2
Fig. 2

Schematics of the relationship between wave-front aberration and ray aberration in the S–H test.

Fig. 3
Fig. 3

S–H grid at point i, where point i is an interior grid point. It is similar to the cross difference scheme of partial differential equations.

Fig. 4
Fig. 4

Extended S–H grid array. Circular domain Ω0 is the Ø500-mm thin mirror (with 6-mm thickness) of the active optics experiment system in the NAIRC. There are 161 S–H grid points in Ω0. After domain extension, Ω0 becomes a 15 × 15 (or 17 × 17) square grid mesh Ω1, which has 225 (or 289) grip points.

Fig. 5
Fig. 5

Compositum Ω0 and the extended Ω0, Ω1, D1, D2, D3, etc., are the blind areas with boundaries C1, C2, C3, etc., in Ω0. C0 is the outer boundary of Ω0, and ∂Ω0 is the outer boundary of Ω0.

Fig. 6
Fig. 6

Point 40 is a corner-boundary grid point in Ω0. Points 40, 41, and 57 are S–H grid points in Ω0; points 23 and point 39 are the invented grid points in Ω1Ω0.

Fig. 7
Fig. 7

Relationship between the relaxation factor ω and the corresponding iteration times for convergence. It indicates that the optimal relaxation factor is 1.881 and the corresponding iteration time for convergence is 111.

Fig. 8
Fig. 8

Situation of adjacent points in one direction.

Fig. 9
Fig. 9

Relationship between the constraint points (zero point of the reconstructed wave front) and the corresponding condition numbers of the normal equation set. Graph (a) is the case t = 15, and graph (b) is the case t = 17. The graphs show that, with the same constraint situation, the condition number of the normal equation set in the case of t = 15 is better than in the case of t = 17.

Fig. 10
Fig. 10

(a) is the scalar phase array of an ideal wave front generated by f(z, y) = z 2 + y 2. (b) is the array of phase difference between the ideal wave front and the calculated wave front determined from the gradients of function f(z, y), with employment of the extension algorithm proposed in this paper.

Fig. 11
Fig. 11

Stability demonstration of domain extension algorithm. Using a group of random gradient errors with Gaussian distributions as the initial values and employing the domain extension algorithm, we obtained the reconstructed wave-front phase array shown in (a), and the corresponding three-dimensional map is shown in (b).

Fig. 12
Fig. 12

Stability demonstration of nonextension algorithm. Using the same group of random gradient errors as in Fig. 11 as the initial values and employing the nonextension algorithm, we obtained the reconstructed wave-front phase array shown in (a), and the corresponding three-dimensional map is shown in (b).

Fig. 13
Fig. 13

Result of calculation with the domain extension algorithm. The reconstructed wave-front phase array is shown in (a). The calculated phase values in the extended domain are discarded, because they are not of interest.

Fig. 14
Fig. 14

Result of calculation with the nonextension algorithm. The reconstructed wave-front phase array is shown in (a), and the corresponding three-dimensional map is shown in (b).

Equations (138)

Equations on this page are rendered with MathJax. Learn more.

lyi=yi1-yi0,lzi=zi1-zi0.
Wyi=lyif0,Wzi=lzif0, i=1, 2,, N,
grad W=Wy i+Wz j=f1y, zi+f2y, zj.
Wy=Wyi1-ya+Wyi+1ya,
Wi+1-Wi=12Wyi+Wyi+1a.
wi-wi-1=12Wyi+Wyi-1a,wi+1-wi=12Wyi+1+Wyia,wi-t-wi=12Wzi-t+Wzia,wi-wi+t=12Wzi+Wzi+ta.
wi+1-wi=gi,i+1,  ikt,  k=1, 2,, t;
wi-wi+t=fi,i+t,  i=1, 2,, m-t,
gi,i+1=12Wyi+1+Wyia,
fi,i+t=12Wzi+Wzi+ta.
w2-w1=g2,1,  w1-w1+t=f1,1+t,w3-w2=g3,2,  w2-w2+t=f2,2+t,  wt-wt-1=gt,t-1,  wt-1-w2t-1=ft-1,2t-1,wt-w2t=ft,2t,wt+2-wt+1=g+2t,t+1,  wt+1-w2t+1=ft+1,2t+1,  wm-t-wm-t-1=gm-t,m-t-1,  wm-t-1-wm-1=fm-t-1,m-1,wm-t-wm=fm-t,m,wm-t+2-wm-t+1=gm-t+2,m-t+1,wm-wm-1=gm,m-1.
AW=F,
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0 0 0 1 w 1 w 2 w t w t+1 w t+2 w 2t w m2t+1 w m2t+2 w mt w mt+1 w m1 w m g 2,1 g 3,2 g t,t+1 g t+2,t+1 g m,m1 f 1,t+1 f 2,t+2 f t,2t f i,i+t f mt1,m1 f mt,m
Di=-11-11-11t-1×t,  i=1, 2,, t.
ATAW=ATF,
ATA=BTB+CTC,
BTB=diagD1TD1, D2TD2,, DtTDtm×m,
DiTDi=1-1-12-1-12-1-11t×t,
CTC=10-110-1-1020-1-1020-1-101-101m×m.
ATA=E1-I-IE2-I-IE2-I-IE1m×m,
E1=2-1-13-1-13-1-12t×t,
E2=3-1-14-1-14-1-13t×t,
-I=-1-1t×t.
rankATA=rankA=m-1.
ATF=BTg+CTf=f1,t+1-g2,1f2,t+2+g2,1-g3,2ft-1,2t-1+gt-1,t-2-gt,t-1ft,2t+gt,t-1ft+1,2t+1-f1,t+1-gt+2,t+1fi,i+1-fi-t,i+gi,i-1-gi+1,ifm,m-t-fm-2t,m-t+gm-t,m-t-1-fm-t,m+gm,m-1.
-wi-t-wi-1+4wi-wi+1-wi-t=fi,i+t-fi-1,i+gi,i-1-gi+1,i,
wi=14wi-t+wi-1+wi+1+wi+t+a8Wyi-1+Wzi+t-Wzi-t-Wyi+1.
w1=12w2+wt+1+a4Wz1+Wzt+1-Wy2-Wy1,
wt=12wt-1+w2t+a4Wzt+Wz2t+Wyt+Wyt-1.
f0y, z=f1y, zi+f2y, zj.
f0y, z=W+ny, z.
 W-f0y, z2dydz
Fy, z, Wy, Wz=W-f0y, z2=Wy i+Wz j-f0y, z2.
JWy, z=ΩFy, z, Wy, Wzdydz.
S=WWC1Ω¯, WnΩ=gy, z.
yFWy+zFWz=0.
2Wy2+2Wz2=i f0y +j f0z .
2W=f0=fy, z,
2W=fy, z,y, zΩ0WnΩ0=gy, z;
WyΩ1\Ω0WzΩ1\Ω00.
w40=14w23+w57+w41+w39+a8Wy39+Wz57-Wz23-Wy41.
Wy39=0,  Wz23=0;
w23-w40=12Wz40+Wz23a=12Wz40a.
w23=w40+12Wz40a.
w40-w39=12Wy40+Wy39a=12Wy40a,
w39=w40-12Wy40a.
w40=12w57+w41+a4Wz40+Wz57-Wy40-Wz41.
ATA=aijm×m,  B=bijm×m=0,i<jbij,ij,
aij=k=1i bikbjk,  1ijm.
BBTW=Y.
U=BTW,  U=u1, u2,, umT.
BU=Y.
u1=y1/b11,ui=yi-j=1i-1 bijyjbii,  i=2, 3,, m.
wm=um/bm,m,wi=ui-j=i+1m biiwjbii,i=m-1, m-2,, 1.
W161=w1, w2,, w161T,pWi33Wy3i, 3Wz3i,  pWi44Wy4i, 4Wz4i,4y4+4z4W|i=Mi4, 3y3+3z3W|i=Mi3.
-wi+1-wi+t+4wi-wi-1-wi-t=-a2ni-a4124Wy4i+4Wz4i.
W=W-W0=w1, w2,, wmT,
N=n1, n2,, nmT,
4y4+4z4W=M14, M24,, Mm4T=M4,
niy, z=nyy, z·i+nzy, z·j;
ni=nyyi+nzzi.
nyyi=ny|ii+1-ny|i-1ia-a24!3nyy3i+Oa4,
nzzi=nz|ii-t-nz|i+tia-a24!3nzz3i+Oa4.
-wi+1-wi+t+4wi-wi-1-wi-t=a-ny|ii+1+ny|i-1i-nz|ii-t+nz|i+ti+a4243nyy3i+3nzz3i-24Wy4i+4Wz4i.
3y3 i+3z3 jN=n13, n23,, nm3=M3.
ATAW=a424M3-2M4+aATN,
X2=XTX1/2=i=1m|xi|21/2
lub2A=maxx0XTATAXXTX1/2=ρATA1/2,
W2a424condATAlub2A2 M3-2M42+a condATA1/2lub2A N2.
N=0,, nNy|1, nNy|2,, nNy|k, 0,, 0,, nNz|1, nNz|2,, nNz|k, 0,2m-tT,
nNy|i=ny|jj+1, nNz|i=nz|j+tj, i=1, 2,, k.
N2=i=1knNy|i2+nNz|i21/2=kσGy2+σGz21/2,
σGy=1ki=1knNy|i21/2,  σGz=1ki=1knNz|i21/2.
σGy=βσCCDyf0,  σGz=βσCCDzf0,
W2=mσw=tσW.
σWa424tcondATAlub2A2 M3-2M42+aβkf0tcondATA1/2lub2A σCCD,
σW0.305a4M3-2M42+8.876 aβf0 σCCD.
σW17.752β/f0σCCD=0.724σCCD.
fz, y=z2+y2.
f/z|i,j=2aif/y|i,j=2aj,  i, j=0, ±1, ±2,, ±7.
f/z|i,j=Rzi, jf/y|i,j=Ryi, j,  i, j=0, ±1, ±2,, ±7,
σerror=σz2+σy21/2=0.160.
σW/σerror8.876a=17.752.
A0=-11-11t-1×t.
A=A0A0A0A0I-II-II-I.
AA0A0A0A0A0A0A00-I0I-I0I-I.
AA0A0A0A0A0A0A00-I00-I00-I.
AA0A00A00A000-I00-I00-I.
AA00000000-I00-I00-I.
rankA=t-1t+t-1=m-1.
wi+1-2wi+wi-1a2=2Wy2i+a2124Wy4i+a43606Wy6i+Oa6,
wi-t-2wi+wi+ta2=2Wz2i+a2124Wz4i+a43606Wz6i+Oa6.
a2Wi=2Wi+a2124Wy4i+4Wz4i+Oa4,
2W=f0=W0+ny, z=2W0+ny, z,
2W-W0=ny, z,
a2Wi=2Wi+a2124Wy4i+4Wz4i+Oa4=ni+a2124Wy4i+4Wz4i+Oa4.
-wi+1-wi+t+4wi-wi-1-wi-t=-a2ni-a4124Wy4i+4Wz4i+Oa6.
wi-wi+t=a Wzi+ti+a3243Wz3i+ti+Oa5.
wi+t=wi-a Wzi+a22!2Wz2i-a33!3Wz3i+a44!4Wz4i+Oa5,
wi-t=wi+a Wzi+a22!2Wz2i+a33!3Wz3i+a44!4Wz4i+Oa5.
wi-t-wi+t2a=Wzi+a23!3Wz3i+Oa4.
wi-wi+t=a Wzi+ti+a3243Wz3i+ti+Oa5.
wi+1-wi=a Wyii+1+a3243Wy3ii+1+Oa5,
Wzi+ti=12Wzi+t+Wzi-a283Wz3i+ti-a43845Wz5i+ti+Oa5.
Wzi+ti=fy, zi+t+a2=fy, zi-a2.
fy, zi+t+a2=fy, zi+t+a2fzi+t+a282fz2i+t+a38×3!3fz3i+t+a416×4!4fz4i+t,
fy, zi-a2=fy, zi-a2fzi+a282fz2i-a38×3!3fz3i+a416×4!4fz4i.
Wzi+ti=fy, zi+t+fy, zi2-a4fzi-fzi+t+a2162fz2i+2fz2i+t-a3963fz3i-3fz3i+t+a47684fz4i+4fz4i+t.
Wzi+ti=12Wzi+Wzi+t-a283Wz3i+ti+Oa3.
Wyii+1=12Wyi+1+Wyi-a283Wy3ii+1-a43845Wy5ii+1+Oa5.
WnΩ1=gy, z=gyi+gzj.
gy=Wy12=-12Wy1+Wy2+a283Wy312+a43845Wy512+Oa5,
gz=Wy1+t1=-12Wz1+Wz1+t+a283Wz31+t1+a43845Wz51+t1+Oa5,
Wy12=w2-w1a-a24!3Wy312+Oa4,
Wz1+t1=w1-w1+ta-a24!3Wz31+t1+Oa4.
-w2+2w1-w1+t=f1,t+1-g1,2+a3123Wy312-3Wz3t+11+a53845Wy512-5Wz5t+11.
-wt-1+2wt-w2t=ft,2t+gt-1,t-a3123Wy3t-1t+3Wz32tt-a53845Wy5t-1t+5Wz52t1t,
-wm-t+2+2wm-t+1-wm-2t+1=-fm-2t+1,m-t+1-gm-t+1,m-t+2+a3123Wy3m-t+1m-t+2+3Wz3m-t+1m-2t+1+a53845Wy5m-t+1m-t+2+5Wz5m-t+11m-2t+1,
-wm-1+2wm-wm-t=-fm-t,m+gm-1,m-a3123Wy3m-1m-3Wz3mm-t-a53845Wy5m-tm-5Wz5m1m-t.
wi+1-2wi+wi-1=a22Wy2i+a4124Wy4i+Oa6.
Wyii+1-Wyi-1i=a2Wy2i+a3244Wy4i+Oa5.
Wyii+1-Wyi-1i=gi,i+1-gi-1,ia-a384Wy4i-a53846Wy6i+Oa6.
a22Wy2i=gi,i+1-gi-1,i-a464Wy4i+Oa6.
wi+1-2wi+wi-1=gi,i+1-gi-1,i-a4124Wy4i+Oa6.
Wni=Wzi+ti=12Wzi+t+Wzi-a283Wz3i+ti-a43845Wz5i+ti.
wi-wi+t=fi,i+t-a3123Wz3i+ti+Oa5.
-wi+1-wi+t+3wi-wi-1=fi,i+t-gi,i+1+gi-1,i-a3123Wz3i+ti+a4124Wy4i+Oa5.
-wi+1-wi-t+3wi-wi-1=-fi,i-t-gi,i+1+gi-1,i+a3123Wz3ii-t+a4124Wy4i+Oa5,
-wi+1-wi-t+3wi-wi+t=-fi-t,i+fi+t,i-gi+1,i+a3123Wy3ii+1+a4124Wz4i+Oa5,
-wi-1-wi-t+3wi-wi+t=-fi-t,i+fi+t,i+gi-1,i-a3123Wy3i-1i+a4124Wz4i+Oa5.
lub2ATA=ρATA21/2=ρATA=lub2 A2.
W=a424ATA-1M3-2M4+a·ATA-1ATN,
W2a424lub2ATA-1|M3-2M42+a lub2ATA-1AT|N2=a424lub2ATA-1lub2ATAlub2ATA M3-2M4+a lub2ATA-1ATlub2Alub2A N2,
condATA=lub2ATA-1lub2ATA
PA=R0,  A=PTR0.
ATA-1=RTR-1=R-1RT-1,ATA-1AT=R-10P.
lub2R=lub2A,  condATA=condR2.
W2a424condATAlub2A2 M3-2M42+a condATA1/2lub2A·N2.

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