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The document describes polynomial interpolation and Lagrange polynomial interpolation. It defines a function f(x) using values at points x0, x1, ..., xn. The Lagrange polynomial is a linear combination of basis polynomials using the function values as coefficients to exactly interpolate the given data points.

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n P(x) = % $ j "k (x # x j ) f (x k ) k =0$ j "k (x k # x j ) !
y = ax + b
2 y = ax + bx + c y = ax + b
5 y = ax + b 2 ! y = ax + bx + c y = ax 4 + bx 3 + cx 3 + dx1 + e !
a b c d e y = ax 4 + bx 3 + cx 3 + dx1 + e f (x1 ) = ax14 + bx13 + cx12 + dx1 + e ! f (x 2 ) = ax 2 4 + bx 2 3 + cx 2 2 + dx 2 + e 4 3 2 f (x 3 ) = ax 3 + bx 3 + cx 3 + dx 3 + e 4 3 2 f (x 4 ) = ax 4 + bx 4 + cx 4 + dx 4 + e f (x 5 ) = ax 5 4 + bx 5 3 + cx 5 2 + dx 5 + e
f (x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) a1 n+1
f (x) = a1 +a2 (x " x1 ) +a3 (x " x1 )(x " x 2 ) ! +a4 (x " x1 )(x " x 2 )(x " x 3 ) +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) !
x=x f (x) = a1 +a2 (x " x1 ) 0 3 f (xx1 ) = a1 +a (x " x )(x " ) 1 2 0 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 ! !
x=x2 f (x) = a1 +a2 (x " x1 ) f (x ) = a + a2 (x 2 " x1 ) 2 1 +a3 (x " x1 )(x " x 2 ) 0 a1 a2 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 !
x=x3 f (x) = a1 +a2 (x " x1 ) f (x ) = a + a (x " x ) + a3 (x 3 " x1)(x 3 " x 2 ) 3+a (x " x 2)(x3" x 1 1 3 1 2) a1 a2 a3 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 !
a1 n+1 f (x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) f (x1 ) = a1 ! f (x 2 ) = a1 + a2 (x 2 " x1 ) f (x 3 ) = a1 + a2 (x 3 " x1 ) + a3 (x 3 " x1 )(x 3 " x 2 ) M f (x n +1 ) = a1 + a2 (x n +1 " x1 ) + a3 (x n +1 " x1 )(x n +1 " x 2 ) +L +an +1 (x n +1 " x1 )(x n +1 " x 2 )(L)(x n +1 " x n )
n f (x) = % $ j "k (x # x j ) f (x k ) k =0 $ j "k (x k # x j ) !
n f (x) = % $ j "k (x # x j ) f (x k ) k =0$ j "k (x k # x j ) (x " x1 )(x " x 2 )(x " x 3 )L(x " x n ) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )L(x 0 " x n ) (x " x 0 )(x " x 2 )(x " x 3 )L(x " x n ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )L(x1 " x n ) +LLLLLLLLLLLLLLLLLL (x " x 0 )(x " x1 )(x " x 2 )L(x " x n "1 ) + f (x n ) (x n " x 0 )(x n " x1 )(x n " x 2 )L(x n " x n "1 )
(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x 2 ) + x (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 )
(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) f(x0) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) 0 (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 5 ) f (x 2 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) 0 (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) f(x1) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 51) f (x 2 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) 0 (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) 0 (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 52) f (x 2 ) f(x2) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
f (x) = f (x 0 ) f (x) = f (x1 ) f (x) = f (x 2 ) f (x) = f (x 3 ) ! f (x) = f (x 4 ) ! f (x) = f (x 5 ) !
P(x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) (x " x1 )(x " x 2 )(x " x 3 )L(x " x n ) P(x) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )L(x 0 " x n ) ! (x " x 0 )(x " x 2 )(x " x 3 )L(x " x n ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )L(x1 " x n ) +LLLLLLLLLLLLLLLLLL (x " x 0 )(x " x1 )(x " x 2 )L(x " x n "1 ) + f (x n ) (x n " x 0 )(x n " x1 )(x n " x 2 )L(x n " x n "1 )
f (x) = ax + b f (x 2 ) " f (x1 ) a " x1 a= x 2 " x1 b = f (x1 ) " a # x1 ! f (x) = ax + f (x1 ) " a # x1 = a(x " x1 ) + f (x1 ) f (x 2 ) " f (x1 ) = (x " x1 ) + f (x1 ) x 2 " x1
f (x 2 ) " f (x1 ) f (x) = (x " x1 ) + f (x1 ) x 2 " x1 x " x1 x " x1 = f (x 2 ) " f (x1 ) + f (x1 ) x 2 " x1 x 2 " x1 x " x1 # x " x1 & = f (x 2 ) + f (x1 )$1 " ' x 2 " x1 % x 2 " x1 ( x " x1 # (x 2 " x1 ) " (x " x1 ) & = f (x 2 ) + f (x1 )$ ' x 2 " x1 % x 2 " x1 ( x2 " x x " x1 = f (x1 ) + f (x 2 ) x 2 " x1 x 2 " x1 !

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Math01_ogashin

  • 1.
  • 2.
  • 3.
  • 4.
  • 5.
  • 6. n P(x) = % $ j "k (x # x j ) f (x k ) k =0$ j "k (x k # x j ) !
  • 7.
  • 8.
  • 9. y = ax + b
  • 10. 2 y = ax + bx + c y = ax + b
  • 11. 5 y = ax + b 2 ! y = ax + bx + c y = ax 4 + bx 3 + cx 3 + dx1 + e !
  • 12. a b c d e y = ax 4 + bx 3 + cx 3 + dx1 + e f (x1 ) = ax14 + bx13 + cx12 + dx1 + e ! f (x 2 ) = ax 2 4 + bx 2 3 + cx 2 2 + dx 2 + e 4 3 2 f (x 3 ) = ax 3 + bx 3 + cx 3 + dx 3 + e 4 3 2 f (x 4 ) = ax 4 + bx 4 + cx 4 + dx 4 + e f (x 5 ) = ax 5 4 + bx 5 3 + cx 5 2 + dx 5 + e
  • 13.
  • 14. f (x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) a1 n+1
  • 15. f (x) = a1 +a2 (x " x1 ) +a3 (x " x1 )(x " x 2 ) ! +a4 (x " x1 )(x " x 2 )(x " x 3 ) +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) !
  • 16. x=x f (x) = a1 +a2 (x " x1 ) 0 3 f (xx1 ) = a1 +a (x " x )(x " ) 1 2 0 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 ! !
  • 17. x=x2 f (x) = a1 +a2 (x " x1 ) f (x ) = a + a2 (x 2 " x1 ) 2 1 +a3 (x " x1 )(x " x 2 ) 0 a1 a2 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 !
  • 18. x=x3 f (x) = a1 +a2 (x " x1 ) f (x ) = a + a (x " x ) + a3 (x 3 " x1)(x 3 " x 2 ) 3+a (x " x 2)(x3" x 1 1 3 1 2) a1 a2 a3 +a4 (x " x1 )(x " x 2 )(x " x 3 ) 0 +a5 (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) 0 !
  • 19. a1 n+1 f (x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) f (x1 ) = a1 ! f (x 2 ) = a1 + a2 (x 2 " x1 ) f (x 3 ) = a1 + a2 (x 3 " x1 ) + a3 (x 3 " x1 )(x 3 " x 2 ) M f (x n +1 ) = a1 + a2 (x n +1 " x1 ) + a3 (x n +1 " x1 )(x n +1 " x 2 ) +L +an +1 (x n +1 " x1 )(x n +1 " x 2 )(L)(x n +1 " x n )
  • 20.
  • 21. n f (x) = % $ j "k (x # x j ) f (x k ) k =0 $ j "k (x k # x j ) !
  • 22.
  • 23.
  • 24. n f (x) = % $ j "k (x # x j ) f (x k ) k =0$ j "k (x k # x j ) (x " x1 )(x " x 2 )(x " x 3 )L(x " x n ) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )L(x 0 " x n ) (x " x 0 )(x " x 2 )(x " x 3 )L(x " x n ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )L(x1 " x n ) +LLLLLLLLLLLLLLLLLL (x " x 0 )(x " x1 )(x " x 2 )L(x " x n "1 ) + f (x n ) (x n " x 0 )(x n " x1 )(x n " x 2 )L(x n " x n "1 )
  • 25. (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x 2 ) + x (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 )
  • 26. (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) f(x0) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) 0 (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 5 ) f (x 2 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
  • 27. (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) 0 (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) f(x1) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 51) f (x 2 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
  • 28. (x " x1 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) f (x) = f (x 0 ) 0 (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )(x 0 " x 4 )(x 0 " x 5 ) (x " x 0 )(x " x 2 )(x " x 3 )(x " x 4 )(x " x 5 ) + f (x1 ) 0 (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )(x1 " x 4 )(x1 " x 5 ) + f (x) = f (x ) (x " x 0 )(x " x1 )(x " x 3 )(x " x 4 )(x " x 5 ) (x 2 " x 0 )(x 2 " x1 )(x 2 " x 3 )(x 2 " x 4 )(x 2 " x 52) f (x 2 ) f(x2) (x " x 0 )(x " x1 )(x " x 2 )(x " x 4 )(x " x 5 ) + f (x 3 ) (x 3 " x 0 )(x 3 " x1 )(x 3 " x 2 )(x 3 " x 4 )(x 3 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 5 ) + f (x 4 ) (x 4 " x 0 )(x 4 " x1 )(x 4 " x 2 )(x 4 " x 3 )(x 4 " x 5 ) (x " x 0 )(x " x1 )(x " x 2 )(x " x 3 )(x " x 4 ) + f (x 5 ) (x 5 " x 0 )(x 5 " x1 )(x 5 " x 2 )(x 5 " x 3 )(x 5 " x 4 ) !
  • 29. f (x) = f (x 0 ) f (x) = f (x1 ) f (x) = f (x 2 ) f (x) = f (x 3 ) ! f (x) = f (x 4 ) ! f (x) = f (x 5 ) !
  • 30. P(x) = a1 + a2 (x " x1 ) + a3 (x " x1 )(x " x 2 ) +L +an +1 (x " x1 )(x " x 2 )(L)(x " x n ) (x " x1 )(x " x 2 )(x " x 3 )L(x " x n ) P(x) = f (x 0 ) (x 0 " x1 )(x 0 " x 2 )(x 0 " x 3 )L(x 0 " x n ) ! (x " x 0 )(x " x 2 )(x " x 3 )L(x " x n ) + f (x1 ) (x1 " x 0 )(x1 " x 2 )(x1 " x 3 )L(x1 " x n ) +LLLLLLLLLLLLLLLLLL (x " x 0 )(x " x1 )(x " x 2 )L(x " x n "1 ) + f (x n ) (x n " x 0 )(x n " x1 )(x n " x 2 )L(x n " x n "1 )
  • 31. f (x) = ax + b f (x 2 ) " f (x1 ) a " x1 a= x 2 " x1 b = f (x1 ) " a # x1 ! f (x) = ax + f (x1 ) " a # x1 = a(x " x1 ) + f (x1 ) f (x 2 ) " f (x1 ) = (x " x1 ) + f (x1 ) x 2 " x1
  • 32. f (x 2 ) " f (x1 ) f (x) = (x " x1 ) + f (x1 ) x 2 " x1 x " x1 x " x1 = f (x 2 ) " f (x1 ) + f (x1 ) x 2 " x1 x 2 " x1 x " x1 # x " x1 & = f (x 2 ) + f (x1 )$1 " ' x 2 " x1 % x 2 " x1 ( x " x1 # (x 2 " x1 ) " (x " x1 ) & = f (x 2 ) + f (x1 )$ ' x 2 " x1 % x 2 " x1 ( x2 " x x " x1 = f (x1 ) + f (x 2 ) x 2 " x1 x 2 " x1 !