About the Project
NIST

minimum

AdvancedHelp

(0.001 seconds)

1—10 of 41 matching pages

1: 3.1 Arithmetics and Error Measures
Let E min E E max with E min < 0 and E max > 0 . …The integers p , E min , and E max are characteristics of the machine. …Underflow (overflow) after computing x 0 occurs when | x | is smaller (larger) than N min ( N max ). … In the case of the normalized binary interchange formats, the representation of data for binary32 (previously single precision) ( N = 32 , p = 24 , E min = - 126 , E max = 127 ), binary64 (previously double precision) ( N = 64 , p = 53 , E min = - 1022 , E max = 1023 ) and binary128 (previously quad precision) ( N = 128 , p = 113 , E min = - 16382 , E max = 16383 ) are as in Figure 3.1.1. … N min x N max , and …
2: 18.37 Classical OP’s in Two or More Variables
18.37.1 R m , n ( α ) ( r e i θ ) = e i ( m - n ) θ r | m - n | P min ( m , n ) ( α , | m - n | ) ( 2 r 2 - 1 ) P min ( m , n ) ( α , | m - n | ) ( 1 ) , r 0 , θ , α > - 1 .
18.37.3 R m , n ( α ) ( z ) = j = 0 min ( m , n ) c j z m - j z ¯ n - j ,
18.37.4 x 2 + y 2 < 1 R m , n ( α ) ( x + i y ) ( x - i y ) m - j ( x + i y ) n - j ( 1 - x 2 - y 2 ) α d x d y = 0 , j = 1 , 2 , , min ( m , n ) ;
18.37.6 R m , n ( α ) ( z ) = j = 0 min ( m , n ) ( - 1 ) j ( α + 1 ) m + n - j ( - m ) j ( - n ) j ( α + 1 ) m ( α + 1 ) n j ! z m - j z ¯ n - j .
3: 15.14 Integrals
15.14.1 0 x s - 1 F ( a , b c ; - x ) d x = Γ ( s ) Γ ( a - s ) Γ ( b - s ) Γ ( a ) Γ ( b ) Γ ( c - s ) , min ( a , b ) > s > 0 .
4: 1.4 Calculus of One Variable
Maxima and Minima
A necessary condition that a differentiable function f ( x ) has a local maximum (minimum) at x = c , that is, f ( x ) f ( c ) , ( f ( x ) f ( c ) ) in a neighborhood c - δ x c + δ ( δ > 0 ) of c , is f ( c ) = 0 . …
§1.4(vii) Maxima and Minima
If f ( x ) is twice-differentiable, and if also f ( x 0 ) = 0 and f ′′ ( x 0 ) < 0 ( > 0 ), then x = x 0 is a local maximum (minimum) (§1.4(iii)) of f ( x ) . The overall maximum (minimum) of f ( x ) on [ a , b ] will either be at a local maximum (minimum) or at one of the end points a or b . …
5: 5.14 Multidimensional Integrals
provided that a , b > 0 , c > - min ( 1 / n , a / ( n - 1 ) , b / ( n - 1 ) ) . … when a > 0 , c > - min ( 1 / n , a / ( n - 1 ) ) . …
6: 20.6 Power Series
20.6.1 | π z | < min | z m , n | ,
where z m , n is given by (20.2.5) and the minimum is for m , n , except m = n = 0 . …
7: 19.27 Asymptotic Approximations and Expansions
19.27.13 R J ( x , y , z , p ) = 3 2 z p ( ln ( 8 z a + g ) - 2 R C ( 1 , p z ) + O ( ( a z + a p ) ln p a ) ) , max ( x , y ) / min ( z , p ) 0 .
19.27.14 R J ( x , y , z , p ) = 3 y z R C ( x , p ) - 6 y z R G ( 0 , y , z ) + O ( x + 2 p y z ) , max ( x , p ) / min ( y , z ) 0 .
19.27.15 R J ( x , y , z , p ) = R J ( 0 , y , z , p ) - 3 x h p ( 1 + O ( ( b h + h p ) x h ) ) , x / min ( y , z , p ) 0 .
8: 19.9 Inequalities
19.9.7 ( 1 - 1 4 k 2 ) - 1 / 2 < 4 π k 2 ( E ( k ) - k 2 K ( k ) ) < min ( ( k ) - 1 / 4 , 4 / π ) ,
19.9.13 Π ( ϕ , α 2 , 0 ) Π ( ϕ , α 2 , k ) min ( Π ( ϕ , α 2 , 0 ) / Δ , Π ( ϕ , α 2 , 1 ) ) .
9: 14.21 Definitions and Basic Properties
The generating function expansions (14.7.19) (with P replaced by P ) and (14.7.22) apply when | h | < min | z ± ( z 2 - 1 ) 1 / 2 | ; (14.7.21) (with P replaced by P ) applies when | h | > max | z ± ( z 2 - 1 ) 1 / 2 | .
10: 19.34 Mutual Inductance of Coaxial Circles
is the square of the maximum (upper signs) or minimum (lower signs) distance between the circles. …