A A A A A B A =, C =. f k k x k l f f 3 = k 3 k 3 x k 34 k 34 x 3 k 3 l 3 k 34 l 34. f 4 x 4 K K = [ ] 4 K = K (K) = (K)
I m m R I m m = [e, e,, e m ] R = [a, a,, a m ] R = (r ij ) r r r m R = r r m. r mm I m m = R R r r r m [e, e,, e m ] = [a, a,, a m ] r r m. r mm R R m i= r ii r ii R e = r a a = r e C m (k) {v = v v Cm : v i = i > k} v m C m () C m () C m (m) = C m C m (k) C m a C m () k i a k C m (k) I m m = RR e i+ = m a k r k(i+) = k= a i+ = r (i+)(i+) ( i a k r k(i+) + a i+ r (i+)(i+). k= e i+ ) m a k r k(i+) C m (i + ). k= a k C m (k) k m R d d d c c F d 8 f j (i) d C 8 c C 8 F c = d F C 8 F {} d c Ad = c c C 8 AF c = c c C 8 A = F A F ij = f j (i) c c 8 A A = A A
m m m x + x = (x i + x i ) = (x i + x i) + x i x i = x + x + x x = x + x, i= i= k i= x i = k i= x i k n n+ i= i= n x i = x i + x n+ = i= n n+ x i + x n+ = x i. n N Ax = λx x C \ {} λ x = λx x = x (λx) = x Ax = x A x = (Ax) x = (λx) x = λ x x = λ x. x λ = λ Ax = λx Ay = µy λ µ λy x = y (Ax) = (y A)x = (y A )x = (Ay) x = (µy) x = µy x, µ = µ λ µ y x = x y A λ A x x λ = i= i= x x = x (A A)x = (Ax) (Ax) = (λx) (λx) = λ x x. A (A + A ) (A A ) is (is) = i S = i( S) = is λ S λi is λ x (I S)x = x = Sx x x = (Sx) x = x S x = x Sx = x x. x x = x (I S) = {} I S
A = [(I S) (I + S)] [(I S) (I + S)] U (U ) = (U ) A = (I + S) (I S) (I S) (I + S) = (I + S )[(I S)(I S) ] (I + S) = (I S)[(I S)(I + S)] (I + S) = (I S)(I S ) (I + S). A = A(I S)(I S) = (I S)[(I S ) (I S )](I S) = (I S)(I S) = I Q u = u A x C \ {} Ax = x + uv x = x = u(v x) u x = αu α C Ax = αu + u(v (αu)) = αu( + v u) =. v u = v u = α \{} A(αu) = αu + uv (αu) = αu( + v u) = A A v u = (A) = {αu : α C} u A A A [x,, x m ] AA = (I + uv )[x,, x m ] = [x + uv x,, x m + uv x m ] = I = [e,, e m ]. x i +uv x i = e i ( i m) A [e θ u,, e m θ m u] = I uθ θ = θ θ m I = AA = (I + uv )(I uθ ) = I uθ + uv uv uθ, = uθ i +uv i u(v u)θ i ( i m) θ i θ i = v i +v u A = I uv +v u [ ] H k+ = Hk H k (H k ) = H k H k [ H k H k H k H k ]. H k α k α = [ ] [ Hk+ T H T = k Hk T Hk H ] = α k k = α k H k+. H T k H T k H k H k H k (k N) α k = k
x W = W x x W = W x = W x = x = W x + y W = W (x + y) = W x + W y W x + W y = x W + y W. αx W = W (αx) = α(w x) = α W x = α x W λ A x C m \ {} Ax = λx λ = Ax x Ay = A. y C m \{} y x i = i m x i x = x i m i= x i = x x = αe i (α C, i m) x i = i m x i x = m i= x i m i= x i = m x i = m x x = α (α C) x C n \ {} Ax Ax x = n Ax. n x x A n A x C n \ {} Ax x m Ax x. A m A A i i A A i i A B A
p u C m v C n E = uv x C n \ {} Ex x = uv x x = v x u x u v, E u v x = v E = u v E ij = u i v j m n m n m n E F = E ij = u i v j = ( u i )( v j ) = u F v F. i= j= i= j= i= j= x x ( x x ) x = x x = x = x + x = y (x + x ) ( y x + y x ) y x + y x = x + x. y = y = y = y = α C αx = y = y (αx) = y = α y x = α x x z C m z x = z x z = e iθ z / z θ = (z x) z = z x = e iθ z z x = z x z = x =. Bx = (yz )x = y(z x) = y B = Bw = yz w = z w y = w z = z =. w = w = w = w = z B = yz Bx = y B = B Bx = y B = z x = = x z = x X l X l(x) = x l = l(x) = l( m i= x ie i ) = [l(e ),, l(e m )] x z z = [l(e ),, l(e m )] l(w) = z w w C m z x = l(x) = x z = l = x m
A = σ Av σ w = v c + v s c s c + s = Aw < σ Aw < σ {σ j } {σj } AA A A {u j } AA σj σ j AA σj AA u j {v j } A A = UΣV A AA = U(ΣΣ )U (AA )U = U(ΣΣ ) U [u, u,, u m ] { (AA [σ )[u, u,, u m ] = u, σu,, σmu m ] m n [σu,, σnu n,,, ] m > n. p = {m, n} σ σ σ p AA u u u p U m n U σ σ σ m m > n n U σ σ σ n m n U AA U A p A V (A A)V = V (ΣΣ ) ΣΣ Σ Σ U = I Σ = U = [ ] Σ = [ ] [ ] 3 V = [ ] 3 V = U = I 3 3 Σ = V = U = I Σ = U = [ ] [ ] [ ] V = [ ] Σ = [ ] [ ] V = [ ] {u, w, v} = [{{, }, {, }}]
A = a a a n B = a m a m, a a m a m a mn a mn a m,n a n B A T A B A B On pre image plane 3 On image plane.8.6.4...4.6.8.5.5 3 3 3
A U Σ V B U Σ V A B Q A = QBQ A = Q(U Σ V )Q = (QU )Σ (QV ) (QU )Σ (QV ) A Σ = Σ A B [ ] [ ] A = B = A B A
C m R m A A A A v v v n V = [v,, v n ] λ i v i (i =,, n) A A λ λ λ n r λ r > λ r+ = = λ n = (A A) = (A) x (A A) A Ax = (Ax) (Ax) = x A Ax = Ax = x (A). σ i = λ i u i = Avi σ i i =,, r {u,, u r } u i u j = (Av i) (Av j ) σ i σ j = v i (A A)v j λi λ j = λ jvi v j = δ ij. λi λ j {u,, u r } R m {u,, u r, u r+,, u m } U = [u,, u m ] U AV (A A) = (A) u { U AV = u A[v,, v n ] = (u i Av j ), u j > r i Av j = σ i δ ij j r. u m U AV Σ A = UΣV A [ ] [ ] AA = = [ ] [ ] 5 4, (λi AA λ 5 4 ) = = λ 9λ + 4. 4 4 4 λ 4 AA λ, = 9± 65 ( 9+ A σ(a) = ) / ( 65.9896648889 σ 9 (A) = ) / 65.6847464898 A UΣV ε > A ε = U(Σ + εi m n )V I ij = δ ij Σ (Σ + εi m n ) A ε A A ε = U(εI m n )V = ε. ε A A ε = A ε C m n
U AA T = UΣΣU T [ ] [ ] AA T 5 75 5 3 = = 5 75 5 3 5 [ ] 5 3 3 5 ([ ]) 5 λ 3 = λ λ + 6 = (λ 8)(λ ). 3 5 λ AA T 8 5 = 5 = 5 A 5 [ ] [ ] 5 3 8 U = U 3 5 U a, b {, } V = A T UΣ = [ ] 5 U = [ a a [ a a ] b, b ] b b [ 5 ] [ 3 = U V a = b = 5 a 4 5 b 4 5 a 3 5 b A 5 [ ] [ ] [ ] [ ] 3/5 4/5 / ± ± ± 4/5 3/5 / / ± / R A ]. On pre image plane 5 On image plane.8.6.4 5.. 5.4.6.8.8.6.4...4.6.8 5 5 5 5 5 A = A F = + 5 = 5 A = 6 A = 5
A = UΣV T U V [ A = (UΣV T ) = V Σ U T 3 = 5 4 5 4 5 3 5 ] [ 5 ] [ ] = [ ] 5. A [ ] λ = λ 3λ +. 5 λ A λ, = (3 ± 39i)/ A = 5 ( ) = λ λ = 4 (9 + 39) = σ σ = 5 = (A) = [ ] A A = = [ ] [ ] [ ] V Σ U Im m UΣV UΣV = I m m V Σ U [ ] [ ] [ ] [ ] Im m U Σ V I m m V Σ U. [ ] Im m I m m [ ] [ ] U Im m V I m m [ ] A A [ ] V U = [ Im m I m m [ ] [ ] U U V V ] [ U V [ ] [ Im m U I m m V ] ] [ ] Im m. I m m ([ ] [ ]) ([ ] [ ]) ([ ] [ Im m U Σ Im m Im m U = I m m V Σ I m m I m m V ([ ] [ ]) [ ] ([ ] [ ]) Im m U Σ Im m U = I m m V Σ. I m m V Σ Σ = Σ [ ] [ ] [ ] [ ] [ ] [ ] Σ Im m I m m Im m I = m m Σ Im m I, m m Im m I m m = I Σ I m m I m m I m m I m m Σ I m m I m m I m m I m m. m m ])
[ ] Σ = Σ X = [ ] Im m I m m I m m I m m [ Σ Σ [ Im m I m m [ ] Σ Σ ] ] [ U V [ ] Σ X X Σ ] [ Im m [ ] Im m I m m I m m I m m I m m [ A A ] I m m I m m ], (I P ) = I P = I P (I P ) (I P ) = [(I P ) P ] = (I P ) (I P )P P (I P ) + P = I P + P = I, I P F = I E = 4 (I + F + F ) = I+F = E E F F = F = F E = E E E E = A A = UΣV A Σ A A = V Σ ΣV A A Σ A A A A A x A Ax = Ax = x A Ax = A A x Ax = A Ax = A A
A A a = / / a = P A P x = a xa + a xa = a a x + a a x = ( ) a / / (a, a ) a x = AA x x C 3 P = AA = / / P (,, 3) (,, ) P = B(B B) B = 5. 6 5 P (,, 3) (,, ) v (P ) P v = v P = x Cm \{} P x x UΣV x C m P = P x = x P P x = x V Σ U UΣV x = x V Σ V x. P x ΣV x Σx = x C m \{} x x C m \{} V = = Σ. x x C m \{} x P P = Σ = Σ = I P = UΣV UΣV = P = UΣV V UΣ = I Σ = I U = V P A A A = ( ) = = B = (b, b ) ( ) r r (b, b ) = (q, q ) r r = b = q = b / b = (/,, / ) T r = q b = (/,, / ) b r q = / / = =
r = 3 B / / 3 B = / ( ) 3 / /. 3 3 B / 6 q 3 = q q = / 6 /. 6 / / 3 / 6 B = / 3 / 6 / / 3 / 3. 6 ˆR r ij = i > j i j i j A A = A A = QR A A = Q R = m j= r jj r ij = qi a j(i j) r jj = a j j i= r ijq i j r jj = a j i= j r ij q i = (a j i= j r ij qi )(a j i= j r ij q i ) = a j i= j rij + i= r ij a j. A = m j= r jj m j= a j m = P P () P () P x () y () x () y () 3 (x (), y () ) q () 3 Q x () y () 3 (x (), y () ) q () 3 Q x () y () v P q () 3 q () 3 Q 3 (q () ) v 3 3, q() 3 A = (a, a,, a n ) ˆQ = (q, q,, q n ) r r r n ˆQ = r r n. r nn
A k k n a k a a k r kk = r kk = k k n k a k = r ik q i a,, a k. i= a k a a k A A n ˆR k a,, a k = q,, q k. A k A k ˆQ ˆR = (q, q, q 3 ) = (q,, q ) = A. (A) = > k = q v = a q = v v m m R m m q m + m = 3m m v v q 3m q q q j q j v j = P qj P q P q a j. P q a j = a j q q a j m (m ) q a j m q (q a j ) m a j q (q a j ) m+(m )+m+m = 4m (j ) v j (4m )(j ) q q j = vj v j 3m q j (4m )(j )+3m n j= [(4m )(j )+3m] = (4m ) n(n ) + 3mn k = a = k = a =
R j R j = = r jj r j(j+) r jj r jn r jj r jj r j(j+) r jn q i = v i /r ii v j = v j r ij q i
slope =.69779 logarithm of max error between the first 4 discrete and continuous Legendre polynomials 4 6 8 4 4 6 8 4 6 8 (base /) power of the grid spacing A A A m A N m m N (m ) N m = A = (I +N) = I N +(N) +( ) m (N) m m = 3 A =, A = 4. σ m A σ m = A. σ m A A e m = (( ) m, ( ) m,, ) A (( ) m, ( ) m,, ) = 4 m 3 σ m 3 4m. m = 3 σ.939.54 3 4m.8 m = 5 σ.47 3 4m
F I qq q λ x x qq x = λx ( λ)x = (q x)q q = {x : q x = } H λ x q x = µq µ ( λ)µ = µ λ = q = {µq : µ C R} x x xe H F = I F = I F = ± a a m F F m i= a i = F m i= a i = m a i p + = {a i : a i =, i m} p = {a i : a i =, i m} { p + + p = m p + p = m, p + = m p = F = m i= a i = F = H H H H (m ) F = F = F F
Q R..36.54.44.3534.56 9.8995 9.4954 9.6975 Q =.77.396.548.44.558.387, R = 3.99 3.9..97.44.558.4 Q R..36.54.684.3577.44.3534.56.38.58 9.8995 9.4954 9.6975 Q =.77.396.548.94.683.44.558.387.3656.498, R = 3.99 3.9..97.44.558.4.539.4695 Q R..36.54.44.3534.56 9.8995 9.4954 9.6975 Q =.77.396.548.44.558.387, R = 3.99 3.9..97.44.558.4 QR = (q,, q n ) Q R r r n ( x F = y) ( ( ) c s x = s c) y ( ) cx + sy. sx + cy
( x y) F ( x y ) [( ) x + F y ( x F y) ( x y )] [( x y ( c x + s y ) ( x y) s + c+ y ) F F ( ( ) ( ) x θ θ r α J = y) θ θ r α J θ ( )] x = [ (x, y) y ( ) x (x, y)f F y ( ) θ α + θ α = r = r θ α + θ α ( )] x =. y ( ) (α θ). (α θ) m = n ( ) m > n A ˆQ ˆR ˆQ = ˆR ˆQ ˆR n n ˆQ n n A = ˆQ ˆR A + = (A A) A = ( ˆR ˆQ ˆQ ˆR) ( ˆR ˆQ ) = ˆR ˆQ = A ˆQ ˆQ. A + A ˆQ ˆQ ˆQ ˆQ ˆQ ( ) m (m n) H ( ˆQ, H) H H H H n (m n) H (m n) (m n) b R m C m ( ) ( ) ˆQ ˆQ b ˆQ H ˆQ b ˆQ H ( ) = ˆQ b ( ˆQ b H b) ˆQ ˆQ A + A ) = ( ˆQ, H b = b. Q m m ( ) Q Q Q =, Q Q Q n n Q n (m n) Q (m n) n Q (m n) (m n) Q Q = Q Q z R n C n ( ) ( ) Q z Q Q z = z Q Q,
a a a 3 [ a e x + a x + a 3 Γ(x) x ] dx F (a, a, a 3 ) = [ a e x + a x + a 3 Γ(x) x ] dx a F (a, a, a 3 ) = a F (a, a, a 3 ) = a 3 F (a, a, a 3 ) = ex dx ex xdx Γ(x)ex dx ex xdx xdx Γ(x) xdx ex Γ(x)dx xγ(x)dx Γ (x)dx a a a 3 = x x Γ(x) x e x x dx dx. dx
A m > n n x
x x x 3 a a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 a a x 4 x 5 x 6 a a a a 3 a 4 a 5 a 6 a 7 a 8 a 9 a a