APPENDIX
Some Matrix Theory
A I . INTRODUCTION
In this appendix we indicate the results in matrix theory that are needed in the rest of the book. Many of the results should be familiar to the reader already; the more basic of these are not proved here. Useful references for matrix theory are Mirsky (1959 , Bellman (1970), and Graybill (1969). Most of the references to the appendix earlier in the text concern results involving matrix factorizations; these are proved here.
A2. DEFINITIONS
A p X q matrix A is a rectangular array of real or complex numbers a , , , a , * , . ,. ,app, written as
so that a,, is the element in the ith row and j t h column. Often A is written as A =(u,,). We will assume throughout this appendix that the elements of a matrix are real, although many of the results stated hold also for complex matrices. If p = q A is called a square matrix of order p. If q = 1 A is a column uector, and if p = 1 A is a row vecfor. If aij=O for i = l , . . . , p , j = 1 ,... ,q, A is called a zero matrix, written A =0, and if p = q, a,, = 1 for i = 1,. . . ,p and aij = 0 for i # j then A is called the identity matrix of order p, written A = I or A = Ip. The diagonul elements of a p X p matrix A are aI I. a22,. . . , app.
512
Aspects of Multivanate Statistical Theow ROBE I. MUlRHEAD
Copyright 8 1982.2WS by John Wiley & Sons. I ~ C .
Definitions 573
The transpose of a p X q matrix A, denoted by A’, is the q X p matrix obtained by interchanging the rows and columns of A, i.e., if A=(a, , ) then A’=(a,,). If A is a square matrix of order p it is called symmetric if A = A’ and skew-symmetric if A = - A’. If A is skew-symmetric then its diagonal elements are zero.
A p X p matrix A having the form
so that all elements below the main diagonal are zero, is called upper- triangular. If all elements above the main diagonal are zero i t is called lower-triangular. Clearly, if A is upper-triangular then A’ is lower-triangular. If A has the form
so that all elements off the main diagonal are zero, i t is called diagonal, and is often written as
A = diag( a, I , . . . , app ).
The sum of two p X q matrices A and B is defined by
A + B = ( a , , + b , , ) .
If A is p X q and B is 4 X r (so that the number of columns of A is equal to the number of rows of B ) then the product of A and B is the p X r matrix defined by
The product of a matrix A by a scalar a is defined by
aA = ( aa,,).
314 Sonre Murrtw Theory
The following properties are elementary, where, if products are involved, it is assumed that these are defined:
A +( - I )A =O (AB) ’= A’A’ (A’)’ = A ( A + B) ’= A‘+ B’ A( B C ) = ( A E ) C
( A + B ) C = AC+ BC A1 = A .
A( B t- C ) = A B + AC
A p X p matrix A is called orrhogonol if AA‘= Ip and idempotent if A’ = A . If A = ( a , , ) is a p X q matrix and we write
A , , = (u , , ) , i - 1, ..., k ; j = l , ..., /
A,, “ ( a , , ) , i = l , . . . , k , j = I + 1 ,..., q
A 2 , = ( u , , ) , i = k + l , . . . , p ; j = l , , . . , /
A,, =(u, , ) , i = k t I ,..., p ; j = I + 1 ,..., q
then A can be expressed as
and is said to bepuriifioned inlo submatrices A, , , A, , , A,, and A,,, Clearly if B is a p X q matrix partitioned similarly to A as
where B , , is k XI, B I Z is k X ( q -/), B,, is ( p - k ) X / a n d B2, is ( p - k ) X ( 9 - /), then
Derermrinanrs 575
Also, if C is a q X r matrix partitioned as
where C,, is / X m , C,, is / X ( r - m ) , C,, is ( 9 - I ) X m , and C,, is ( 9 - / ) X ( r - m), then it is readily verified that
A3. DETERMINANTS
The dererminanf of a square p X p matrix A , denoted by det A or / A [ , is defined by
det A = E,,alJ,aZJ2.. . .,aPJr n
where C, denotes the summation over all p! permutations R = ( j , , . . . , J p ) of ( I , . . . , p ) and en = + 1 or - 1 according as the permutation n is even or odd. The following are elementary properties of determinants which follow readily from the definition:
(i) If every element of a row (or column) of A is zero then det A =O.
(ii) det A =det A’.
(iii) If all the elements in any row (or column) of A are multiplied by a scalar a the determinant is multiplied by a.
(iv) det(aA)=aPdet A.
(v) If B is the matrix obtained from A by interchanging any two of its rows (or columns), then det B = -det A.
(vi) If two rows (or columns) of A are identical, then det A =O.
(vii) If
b , , + c , , b , , + c , , , ..*, b , , + C l ,
aPP I* A = [ a22 9 ...,
u p 2 1 ... , I
so that every element in the first row of A is a sum of two scalars,
576 Sonie Murrix Theory
then
A similar result holds for any row (or column). Hence i f every element in ith row (or column) of A is the sum of n t e rm then det A can be written as the sum of n determinants.
(viii) If B is the matrix obtained from A by adding to the elements of its i th row (or column) a scalar multiple of the corresponding ele- ments of another row (or column) then det B =det A.
The result given in the following theorem is extremely useful.
THEOREM A3.1. If A and B are both p X p matrices then
det(AA)=(det A)(det B )
Proo/. From the definition
where B ( k l , ..., k p ) denotes the p X p matrix whose ith row is the k,th row of B. By property (vi) d e t B ( k , ,..., k,)=O i f any two of the integers
Derermmunrs 511
k,, . . . , k, are equal, and hence
P P I'
det(AB)= 2 - . . 2 detB(k ,,..., k,) k l = l k , = I
k , # k , # . . ' Zk,
By property (v) it follows that
det B( k , ,. . . , k p ) = E,det B ,
where E, = + 1 or - 1 according as the permutation a = ( k , , . . . ,kp) of (1,. , . , p ) is even or odd. Hence
det(AB)= XE,,( fi u i k , ) .det B , r = l
= (det A )(det B ) .
A number of useful results are direct consequences of this theorem.
THEOREM A3.2. I f A , , ..., A,, are all p X p matrices then
det( A I A 2.. .A , ) = (det A , )(det A ) . , . (det A ,, ) .
This is easily proved by induction on n.
THEOREM A3.3. If A is p X p , det(AA')?O.
This follows from Theorem A3.1 and property (ii).
THEOREM A3.4. 9 X q then
If A , , is p X p , A, , is p X q, A,, is q X p , and A,, is
det [ A" ''1 = det [ A" A" ] = (det A, , )(det A 1, 22 A21 22
Proo/. It is easily shown that
det[ '' 0 A22 ]=detA2,
anu
det[ :y]=det A , , .
578 Some Mutrix Theory
Then from Theorem A3. I ,
det[ :::]=det[ I’ O A :,ldet[ ’dl :y]=(det Al,)(det A Z 2 ) .
Similarly
det[ i l l A” ] =det[ 11 det[ ,‘. ] = (dct All)(det A 2 2 ) , 21 22 2 1 A 2 1
THEOREM A3.5. If A is p X 9 and B is 9 X p then
det( Ip + A B ) =det( I9 + BA) .
Prooj We. can write
I,,+ A B A A IP [ 0 I q ] = [ ? B 1 9 ] [ B
so that
( 1 )
Similarly
det( lp + A B ) =det [ JB 4.
so that
det (Iq + BA) = det [-I.. 4- Equating ( I ) and (2) gives the desired result,
Two additional results about determinants are used often.
(ix) If T is m X m triangular (upper or lower) then det T=ny!&.
(x) If H is an orthogonal matrix then det H = t 1.
litverse of a Marnx 579
A4. MINORS AND COFACTORS
If A =(a,,) is a p X p matrix the minor of the element aIJ is the determinant of the matrix MI, obtained from A by removing the ith row andjth column. The cojucfor of a,,, denoted by ail’ is
alJ =( - I)l+’det MIJ.
I t is proved in many matrix theory texts that det A is equal to the sum of the products obtained by multiplying each element of a row (or column) by its cofactor, i.e.,
P
detA= 2 aIJcu,, ( i = l , . . . ,p) J = I
A principal minor of A is the determinant of a matrix obtained from A by removing certain rows and the same numbered columns of A. In general, if A is a p X q matrix an r-square minor of A is a determinant of an r X r matrix obtained from A by removing p - r rows and q-r columns.
A 5 . INVERSE OF A MATRIX
If A =(a,,) is p X p, with det A fO, A is calied a nunsingular matrix. In this case there is a unique matrix B such that A B = Zp. The i - j t h element of B is given by
cuJJ
det A ’ blJ = -
where aJ, is the cofactor of aJi. The matrix B is called the inoerse of A and is denoted by A - I . The following basic results hold:
(i) AA - I = A - I A = I.
(ii) ( A - I)’ = (A’) - I.
(iii) If A and Care nonsingularp X p matrices then ( A C ) - ’ = C-’A- ’ .
(iv) det(A-’)=(det A ) - ’ .
(v) If A is an orthogonal matrix, A - ’ = A‘.
580 Some Mutnx Theory
(vi) If A = diag( u , I,. . . , upp ) with a,, # 0 ( i = 1,. . . , p ) then A - = diag( a;', . . . , u;;).
(vii) If T is an rn X m upper-triangular nonsingular matrix then T-I is upper-triangular and its diagonal elements are I,; I, i = 1,. . . , in .
The following result is occasionally useful.
THEOREM A5 I . Let A and B be nonsingular p X p and y X y matrices, respectively, and let C be p X y and D be q X p. Put P = A + CBD. Then
Prooj Premultiplying the right side of ( I ) by P gives
( A + CBD) [ A - I - A - 'CB( B + BDA - ICB)-'BDA - I] =I-CB(E+ BDA-'CB)~~'EDA~'+CEDA-'
- CBDA- ICB( B + BDA - I c B ) - ' BDA - I
= I + CB [ E - I - ( I + DA - 'CB ) ( B + EDA - 'CB ) - I ] EDA -
=l+CB[B- I - B - l ( B C BDA-'CB)(B+BDA-'CO)-']BDA-'
= I ,
completing the proof.
matrix A in terms of the submatrices of A . The next theorem gives the elements of the inverse of a partitioned
THEOREM A5.2. Partition A and B as
Let A be a p X p nonsingular matrix, and let B = A-I.
whereA,, and B , , are k x k , A,, and B,, are k X ( p - k ) , A,, and B,, are ( p - k ) X k and A,, is ( p - k ) X ( p - k ) ; assume that A,, and A,, are nonsingular. Put
Inverse of a Mutrtx 581
Then
B , , =A,. ’ , , B,,=A,!,, B ,2= -A;1A1 ,A; ! ( ,
B z l = - A,?1,IA,f2.
Proof. The equation AB = I leads to the following equations:
From (6) we have BZl = - A ~ 1 A 2 1 B l l and substituting this in (4) gives A I I B , , - A , , A ~ ’ A , , B , , = I so that B , , = A;; !2 . From ( 5 ) we have B,, = - A ~ ’ A l , B 2 2 r which when substituted in (7) gives A,, B,, - A,,A,’A,,B,, = 1 so that BZ2 = A&’,,.
The determinant of a partitioned matrix is given in the following theo- rem.
THEOREM A5.3. Let A be partitioned as in ( I ) and let A , , . , and A, , I
be given by (3).
(a) If A,, is nonsingular then
det A =det A,,det
(b) If A , , is nonsingular then
det A =det A,,det A,,.,
Proof. To prove (a) note that if
then
502 Some Murnx Theory
(This was demonstrated in Theorem 1.2.10.) Hence
det(CAC’) =(det C)(det A)(det C’)=det A =det A , I . ,det A,, ,
where we have used Theorems A3.2 and A3.4. The proof of (6) is similar.
A6. R A N K OF A MATRIX
If A is a nonzero p X q matrix it is said to have rank r , written rank( A)= r , i f at least one of its r-square minors is different from zero while every ( r + I)-square minor (if any) is zero. If A =O it is said to have rank 0. Clearly if A is a nonsingular p X p matrix, rank(A)= p . The following properties can be readily established:
(i) rank( A)=rank( A’). (ii) If A is p X q, rank(A)smin( p , q).
(iii) If A is p X 4, B is q X r, then
rank( AB)lmin[rank( A),rank( B ) ] ,
(iv) I f A and B are p X q, then
rank( A + B)srank(A)+rank( B).
(v) I f A is P X P, B is p X (7, C is q X q, and A and C are nonsingular, then
rank( A BC) = rank( 8 ) .
(vi) If A is p X 4 and B is q X r such that AB =0, then
rank( B ) 5 q -rank(A).
A7. LATENT ROOTS A N D L A T E N T VECTORS
For a p X p matrix A the chamferistic equarion of A is given by
( 1 ) det( A - A I p ) = O .
The left side of ( I ) is a polynomial of degree p in h so that this equation has exactly p roots, called the latent roots (or characteristic roots or eigenvalues)
I-urent ROOIS md Latent Veclors 583
of A. These roots are not necessarily distinct and may be real, or complex, or both. If X i is a latent root of A then
det(A-X,l)=O
so that A - A l l is singular. Hence there is a nonzero vector x, such that ( A - A, I ) x , = 0, called a latent vector (or characteristic vector or eigenvector) of A corresponding to A,. The following three theorems summarize some very basic results about latent roots and vectors.
THEOREM A7.1. If B = CAC-', where A, B and C are all p X p, then A and B have the same latent roots.
Prooj Since
we have
det(E-AI)=detCdet(A-Al)detCL'=det(A-hl)
so that A and E have the same characteristic equation.
THEOREM A7.2. If A is a real symmetric matrix then its latent roots are all real.
Proof: Suppose that a + ifl is a complex latent root of A, and put
B = [ ( u + i p ) I - ~][(a-ip)l - A]=(*]- A > ~ +P'I .
E is real, and singular because (a + $ ) I - A is singular. Hence there is a nonzero real vector x such that B x = O and consequently
O=x'Ex =x'( al - A)'x + f12x'x
= x'( al - A)'( al - A ) x + PZX'X.
Since x'(a1- A) ' (a l - A)x>O and x'x>O we must have /3 =0, which means that no latent roots of A are complex.
THEOREM A7.3. If A is a real symmetric matrix and A, and A, are two distinct latent roots of A then the corresponding latent vectors x, and x, are orthogonal.
584 Some Mutrix Theory
Proot Since
Ax, = A , K , , Axj =A,xJr
i t follows that
x; Ax, = A,x:x,, x: Ax, = A,x:x,.
Hence ( A , - A,)x:x, ”0, so that x;x, =O. Some other properties of latent roots and vectors are now summarized.
(i) The latent roots of A and A’ are the same.
(ii) If A has latent roots A , , . , . , A p then A - k / has latent roots A , - k , . . . ,Ap - k and kA has latent roots kA, , . . ., kA,.
(iii) If A=diag(u, ,..., a , ) then al, ..., up are the latent roots of A and the vectors (l,O,. . . ,O) , (0, I , . . .,O),. . .,(O,O,. . ., 1) are associated latent vectors.
(iv) If A and R are p X p and A is nonsingular then the latent roots of A B and RA are the same.
(v) I f A, , ..., A, are the latent roots of the nonsingular matrix A then
(vi) If A is an orthogonal matrix ( A N = I ) then all its latent roots have absolute value I .
(vii) If A is symmetric i t is idempotent ( A 2 = A ) i f and only if its latent roots are 0’s and 1’s.
(viii) I f A isp X q the nonzero latent roots of AA’ and A’A are the same.
(ix) If T is triangular (upper or lower) then the latent roots of 7 are the diagonal elements.
(x) I f A has a latent root A of multiplicity r there exist r orthogonal latent vectors corresponding to A. The set of linear combinations of these vectors is called the lutent space corresponding to A. If A, and Aj are two different iatent roots their corresponding latent spaces arc orthogonal.
An expression for the characteristic polynomial p ( A ) = det( A - A Ip) can be obtained in terms of the principal minors of A. Let A,l , lz , , , , , lk be the k X k matrix formed from A by deleting all but rows and columns numbered I , , . . . , i , , and define the k th trace of A as
A - - l , ,..., Ap’are the latent rootsofA-I.
trk(A)= ~ l s , ,<12 . . .< l*~Pde t ,..., 1;
Posrrroe Definite Mutrices 585
The first trace (k = 1) is called the trace, denoted by tr(A), so that tr(A)=Zf= ,a , , . This function has the elementary properties that tr(A)= tr(A’) and if C is p X q, D is q X p then tr(CD)=tr(DC). Note also that tr,,,(A)=det(A). Using basic properties of determinants i t can be readily established that:
(xi) p(A)=det(A - XZ,)=C[=,(- A)’tr,-,(A) [tr,(A)= I]. Let A have latent roots A , , . . . , A p so that
p(A) = (-1)J’ 2 (A - hi). i = I
Expanding this product gives
where 5( A ,, . . . , A p ) denotes the j th elementary symmetric funclioti of h ,,..., A,, given by
(xii) p ( A )= Z[=,( - A )‘rp- k ( A . . ,A, 1.
r ( A , ,.. . , A p ) = 2 ~ l l h 2 . . . ~ l , *
I s i I< r l< . . . < r , S p
Equating coefficients of Ak in (xi) and (xii) shows that
(xiii) r k ( h l , . . . , A p ) = trk(A). It is worth noting that p ( h ) can also be written as
p ( X ) = ( - X)’det Adet(A-’ - A - I I )
P
k = Q =(-X)’detA z ( - A - l ) k t r p - k ( A - l )
and equating coefficients of hk here and in (xii) gives
(xiv) trk( A -‘)=det A - I trp-k( A) .
A8. POSITIVE DEFINITE MATRICES
A p X p symmetric matrix A is called positive (negative) definite if x’Ax>O (KO) for all vectors x f 0; this is commonly expressed as A >O ( A KO). It is called positive (negative) semidefinite if x ’ A x 2 O (SO) for all x Z 0 , written as A 20 ( S O ) . I t is called non-negutioe definite if A >O or A 20, i.e., if x’Ax>O for all x, and non-positive definite if A <O or A SO.
We now summarize some well-known properties about positive definite matrices.
586 Some Mufnx Theory
( i ) A is positive definite i f and only if det A l , > O for i = 1 ,..., p , where A,, , , , , , is the i X i matrix consisting of the first i rows and columns of A.
(ii) If A >O then A- '>O.
(iii) A symmetric matrix is positive definite (non-negative definite) if and only if all of its latent roots are positive (non-negative).
(iv) For any matrix B, BB'rO.
(v) If A is non-negative definite then A is nonsingular if and only if A >O.
(vi) I f A > O i s p X p a n d B i s q X p ( 9 S p ) o f r a n k r then BAB'>OiP r = q and BAB'ZO i f r C q.
(vii) I f A 10, B > O , A - B >O then B - - A - I > O and det A >det B. (viii) If A XI and B>O then det(A + B ) r d e t A +det B.
(ix) If A 1 0 and
whereA,, isa squarematrix, thenA,, >O and A , , - A,2A,1A,l >O.
A9. SOME M A T R I X FACTORIZATIONS
Before looking at matrix factorizations we recall the Gram-Schmidt ortho- gonalization process which enables us to construct an orthonormal basis of R"' given any other basis xI,x2, ..., xm of R". We define
Y I = X I
4 x 2 YiY,
Y;x, y;x, y3 =x3 - I Y 2 - --yt
Y2Y2 Y i Y I
yz =x2 -- --y1
......... ................. ..
and put z, =[ l/(y,'yi)'/2]yl, with i = 1,. .. ,m. Then z,,. . . ,z, form an ortho- normal basis for Rm. Our first matrix factorization utilizes this process.
Some Matrix Factorizutrons 587
THEOREM A9.1. If A is a real m X m matrix with real latent roots then there exists an orthogonal matrix H such that H'AH is an upper-triangular matrix whose diagonal elements are the latent roots of A.
Let A,, ..., Am be the latent roots of A and let x t be a latent vector of A corresponding to A,. This is real since the latent roots are real. Let x 2 ,..., x, be any other vectors such that x l r x 2 ,..., x, form a basis for Rm. Using the Gram-Schmidt orthogonalization process, construct from x I , . . . , x, an orthonormal basis given as the columns of the orthogonal matrix HI, where the first column h, is proportional to x, , so that h, is also a latent vector of A corresponding to A, . Then the first column of AH, is Ah, = X,h,, and hence the first column of H i A H , ish,H;h,. Since this is the first column of A I H ; H l = A , I , , i t is (A,,O, ..., 0)'. Hence
Proot
where A , is (m - I ) X ( m - 1). Since
det(A - A I ) = ( A , -A)det(A, - A l )
and A and H ; A H , have the same latent roots, the lalent roots of A, are A 2 , ..., Am.
Now, using a construction similar to that above, find an orthogonal (m- l )X(m-1) matrix H2 whose first column is a latent vector of A, corresponding to A *. Then
where A, is ( m - 2 ) x ( m - 2 ) with latent roots h3 , . . . ,Xm.
ort hogonai matrix Repeating this procedure an additional m -3 times we now define the
and note that H A H is upper-triangular with diagonal elements equal to A , , . ..'A,,.
An immediate consequence of this theorem is given next.
588 Sonre Mutrrx Theory
THEOREM A9.2. if A is a real symmetric m X m matrix with latent roots A l , . . , , A m there exists an orthogonal rn X m matrix H such that
(2) H’AH = D = diag( A,, . . . , A m ) .
If H =[hl,. . . , h,] then Ir, is a latent vector of A corresponding to the latent root A,. Moreover, if Al, ..., An, are all distinct the representation (2) is unique up to sign changes in the first row of H.
Proof: As in the proof of Theorem A9.1 there exists an orthogonal m X m matrix H I such that
N;AN, = [: ::I- where A*, ..., A, are the latent roots of A , . Since H i A H , is symmetric i t follows that B , =O. Similarly each B, in the proof of Theorem A9.I is zero ( i = I , ..., m - I), and hence the matrix H given by ( I ) satisfies H‘AH- diag(Al, ..., A,,,). Consequently, A h , =A,h, so that 11, is a latent vector of A corresponding to the latent root A,. Now suppose that we also have Q’AQ= D for a orthogonal matrix Q. ‘Then PI)= DP with P = Q’If. If P =( p,,) i t follows that pIJA, = plJA, and, since A , # A,, p , , = O for i f J.
Since P is orthogonal i t must then have the form P = diag( * 1, -L 1 , . . . , -C I ) , and H = QP. THEOREM A9.3. If A is a non-negative definite m X m matrix then there exists a non-negntive definite m X nt matrix, written as such that A = ~ 1 / 2 ~ 1 / 2 .
Proof: Let H be an orthogonal matrix such that H’AH= D, where D=diag(A,, ..., A,) with A l , . . , , A m being the latent roots of A . Since A is non-negative definite, A , 2 0 for i = 1, . . . , m . Putting D112 = diag(A’/2,...,Alm/2), we have D t / 2 D 1 / 2 = D. Now define the matrix At’’’ by A’/* = HD1/211’. Then A ’ / , is non-negative definite and
~ 1 / 2 ~ 1 / 2 = I I D ~ / ~ H ’ ~ I D ~ / ~ I ~ ’ = H D I / ~ D V ~ ~ ~ ’ = H D H ‘ = A .
The term A’/’ in Theorem A9.3 is called a non-negative definite square root of A. If A is positive definite A ‘ / , is positive definite and is called the positive definite square root of A.
THEOREM A9.4. If A is an m X m non-negative definite matrix of rank r then :
(i) There exists an m X r matrix B of rank r such that A = BB’.
Some Mutrix Fuctoriturtons 589
(ii) There exists an m X m nonsingular matrix C such that
A = C [ '1 C'. 0 0
Pro05 As for statement (i), let D, =diag(A,,...,A,) where Al,.,.,Ar are the nonzero latent roots of A, and let H be an m X m orthogonal matrix such that H'AH = diag( A,, . . . ,A,,O,. . , ,O) . Partition H as H =[ H I : H 2 ] , where HI is m X r and H , is m X( m - r); then
Putting DI/' =diag(h'(2,...,X'/2), we then have
where B = H I D:/' is m X r of rank r.
columns are the columns of the matrix B in (i). Then As for statement (ii), let C be an m X ni nonsingular matrix whose first r
The following theorem, from Vinograd (1950). is used often in the text.
THEOREM A9.5. Suppose that A and B are real matrices, where A is k X in and B is k X n , with m I n . Then AA'= BB' if and only if there exists an m X n matrix H with HH'= I,,, such that A H = B.
Proo/. First suppose there exists an m X n matrix H with HH'= I,?# such
Now suppose that AA'= BB'. Let C be a k X k nonsingular matrix such that A H = B. Then BB'= AHH'A'= AA'.
that
[5 #' AA'= BB'= C
(Theorem A9.4), where rank (AA')= r. Now put D = C - ' A , E = C - ' B and
590 Some Murrrx 7heoy
partition these as
where D , is r X m , U, is ( k - r ) X m , E l is r X n , and E, is ( k - r ) X n . Then
and
which imply that E l E ; = D , D ; = I , and 0, =O, E, = O , so that
Now let E2 be an ( 1 1 - r ) X n matrix such that
is an n X n orthogonal matrix, and choose an ( n - r ) X m matrix 6, and an ( n - r ) X ( n - m ) matrix b3 such that
is an n X n orthogonal matrix. Then
and
Some Morrix Fucrorrzarinns 59 I
and hence
E = [ D : 01 fi'E = [ D : O ] Q ,
where Q = D'g is n X n orthogonal. Partitioning Q as
where His m x n and P is (n - m) x n, we then have HH' = I,,, and
c - 1 ~ = E = D H ~ c - 1 ~ ~
so that B = AH, completing the proof.
The next result is an immediate consequence of Theorem A9.5.
THEOREM A9.6. Let A be an n X m real matrix of rank m ( n r m ) . Then :
(i) A can be written as A = H I B, where H , is n X m with H ; H , = In, and B is m X m positive definite.
(ii) A can be written as
where H is n X n orthogonal and B is m X m positive definite.
Proof: As for statement (i), let B be the positive definite square root of the positive definite niatrix A'A (see Theorem A9.3), so that
A'A = B 2 = B'B.
By Theorem A9.5 A can be written as A = H,B, where H , is n X m with
As for statement (ii), let H I be the matrix in (i) such that A = H , B and choose an n X ( n - m ) matrix H2 so that H = [ H , : H 2 J is n X n orthogonal. Then
H ; H , = I,.
We now turn to decompositions of positive definite matrices in terms of triangular matrices.
592 Same Murrrx I’hearv
THEOREM A9.7. If A is an m X 1 ) ~ positive definite matrix then there exists a unique m X m upper-triangular matrix T with positive diagonal elements such that A = T’T.
An induction proof can easily be constructed. The stated result holds trivially for m = 1. Suppose the result holds for positive definite matrices of size m - I. Partition the m X m matrix A as
Proof:
where A , , is ( m - I ) X ( m - I). By the induction hypothesis therc exists a unique ( m - I )X(m - I ) upper-triangular matrix T , , with positive diagonal elements such that A , , = T;,T, , . Now suppose that
where x is ( m - 1)X 1 and y E HI. For this to hold we must have x = (Til)-1a,2, and thcn
- - I y 2 = a 2 2 - x ’ x = a 2 2 - a ’ , 2 T ; ; 1 ( T ~ , ) a12==a22 -a’,2A;;1a,2.
Note that this is positive by (ix) of Section A8, and the unique satisfying this i sy =(a22 -a’,2A,Iw,2)1/2.
Y’O
THEOREM A9.8. If A is an n X m real matrix of rank m ( t i 2 m ) then A can be uniquely written as A = HIT, where H I is n X m with H ; H , = l,,, and T is m X m upper-triangular with positive diagonal elements.
Since A’A is m X m positive definite it follows from Theorem A9.7 that there exists a unique m X m upper-triangular matrix with positive diagonal elements such that A’A = T’T. By Theorem A9.5 there exists an n X m matrix H I with H;tl, = I,,, such that A = /f,T. Note that HI is unique because T is unique and rank( T ) = M.
THEOREM A9.9. I f A is an m X M positive definite matrix and B is an m X m symmetric matrix there exists an m X m nonsingular matrix L such that A = LL’ and tl= LDL’, where D=diag(dl ,..., d,,,), with d, ,..., d,,, being the latent roots of A-IB. If B is positive definite and Jlr. . . ,d, , , are all distinct, L is unique. up to sign changes in the first row of L.
Let A ’ / * be the positive definite square root of A (see Theorem A9.3). so that A = A 1 / 2 A 1 / 2 . By Theorern A9.2 there exists an m X m
Prooj:
Proof:
Some Mumx Fuctorrzurtotrs 593
orthogonal matrix H such that A - ’ / 2 B A - ‘ / 2 = HDH’, where D = diag(d,, ,..,d,,,). Putting L = A’/’H, we now have LL‘= A and B = LDL‘. Note that d ,,..., d, are the latent roots of A-IB.
Now suppose that B is positive definite and the d, are all distinct. Assume that as well as A = LL’ and B = LDL’ we also have A = MM’ and B = MDM’, where M is m X m nonsingular. Then (M- ‘L)( M - ‘L)’=
M - ‘L is orthogonal and QD = DQ. If Q =(9, , ) we then have q,,d, = q,,d, so that q,, =O for i # j . Since Q is orthogonal it must then have the form Q =diag(? I , 2 1,. . ., 2 l), and L = MQ.
THEOREM A9.10. If A is an m X n real matrix (m 5 n ) there exist an m X m orthogonal matrix H and an n X n orthogonal matrix Q such that
M- ‘LLfM- I / = M- ‘ A M - It = M- ‘MM’M’ - I - - 1, so that the matrix Q =
where d, 2 0 for i = 1,. . . , m and d:,. . .,d:, are the latent roots of A X .
Let H be an orthogonal m X m matrix such that AA’= H’D2N, where D 2 =diag(d: ,..., d i ) , with 6,220 for i = 1 ,..., m because AA‘ is non-negative definite. Let D =diag(d,, . . . ,dm) with d, 20 for i = 1,. . . ,111;
then AA’=(H‘D)(H’D)’, and by Theorem A9.5 there exists an m X n matrix Qt with Q,Q; = 1, such that A = H’DQ,. Choose an ( n - m ) X n matrix Q2 so that the n X n matrix
Proo/.
Q = [ Q! Q2
is orthogonal; we now have
A = H’DQ, = H ’ [ D : O ] Q
so that HAQ’=[D:O], and the proof is complete, The final result given here is not a factorization theorem but gives a
representation for a proper orthogonal matrix H (i.e., det H = I ) in terms of a skew-symmetric matrix. The result is used in Theorem 9.5.2.
THEOREM A9. I 1. If H is a proper m X m orthogonal matrix (det H = 1) then there exists an m X m skew-symmetric U such that
1 1 2! H =exp(U) zz I + I/ + - U * + 3 U 3 + . . - .
594 Some Murrix Theory
Proo/. Suppose H is a proper orthogonal matrix of odd size, say, m =2k f 1; it can then be expressed as
H = Q ’
cos8, -sine,
sin 8 , cos 8, 0
cos -sin 8,
sin 8, cos 8, 0
0
0 0 0 0 I . .
0
0 0 0
0
cos 0, -sin 8, 0 sin 8, cos 8, 0 0 0 1
Q,
where Q is rn X m orthogonal and -. n < 8, I n , with i = 1 , . . . , k (see, e.g., Bellman, 1970,p.65). (I f m = 2 k , the last row and column are deleted.) Putting
o - 8 , 0 8, 0 0 0
o -e2 0 82 0 0
o -ek o ek o o
0
0 0 0 0 ’ * . 0 0 0
we then have 0 = exp( Q ’ H Q ) = exp( U ), where U = Q’HQ is skew-symmetric.
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0.025
0.010
0.005
I 1.685
2 1.358
3 1.237
4 1.173
5 1.133
6 1.106
7 1.087
8 1.073
9 1.062
10
1.054
12
1.041
14
1.033
16 1.027
18
1.022
20
1.019
24
1.014
30
1.009
40
1.006
60
1.003
120
1.001
00
1.OOo
x:, 43.745
1.754
1.821
1.385
1.410
1.252
1.266
1.183
1.192
1.140
1.147
1.112
1.117
1.092
1.096
1.077
1.080
1.065
1.068
1.056
1.059
1.043
1.045
1.034
1.036
1.028
1.029
1.023
1.024
1.020
1.020
1.014
1.015
1.010
1.010
1.00
6 1.
006
1.003
1.003
1.001
1.001
1.m
1.OOo
47.400 50.725
1.907
1.969
1.442
1.466
1.284
1.297
1.204
1.213
1.156
1.162
1.124
1.128
1.101
1.105
1.084
1.087
1.072
1.074
1.062
1.06
4
1.047
1-049
1.037
1.039
1.030
1.031
1.025
1.026
1.021
1.022
1.016
1.016
1.011
1.011
1.007
1.007
1.003
1.003
1.001
1.001
1.OOo
1.OOo
54.776 57.648
1.718
1.382
1.256
1.188
1.146
1.117
1.097
1.08 1
1 .069
1.06
0
1.04
6 I .037
1.030
1.025
I .02 1
1.016
1.011
1.007
1.003
1.001
1 .OOo
1.791
1.410
1.272
1.199
1.154
1.123
1.101
1.085
1.073
1.063
,048
,039
.032
.026
.022
.017
1.01 1
1.007
1.003
1 .001
1 .OOo
1.86
0 1.437
1.287
1.209
1.161
1.129
1.106
I .089
1.076
1.066
1.050
1.04
0 1.033
1.027
1.023
1.017
1.012
I .007
1.004
1.001
1 .ooo
1.949
1.410
1.30
6 1.22 1
1.170
1.136
1.11
1 1.093
1 .080
1.069
1.053
1.042
1.034
1.029
1.024
1.018
1.012
I .008
1.004
1.001
1 .Ooo
2.013
1.495
1.319
I .230
1.176
1.141
1.115
I .097
1.082
1.07 1
1.05
4 I .043
1.035
1.029
1.025
1.019
1.013
1.008
1.00
4 1.001
1 .Ooo
47.2122
50.9985
54.4373
58.6192
61.5812
1 1.
750
1.82
4 1.
8%
1.98
8 2.
055
2 1.
405
1.43
4 1.
462
1.49
7 1.
522
3 1.
274
1.29
1 1.
306
1.32
6 1.
340
4 1.
203
1.21
4 1.
225
1.23
8 1.
247
5 1.
158
1.16
7 1.
174
1.18
4 1.
191
6 1.
128
1.13
4 1,
140
1.14
8 1.
153
7 1.
106
1.11
1 1.
116
1.12
2 1.
126
8 1.
089
1.09
4 1.
098
1.10
2 1.
106
9 1.
076
1.08
0 1.
083
1.08
8 1.
090
10
1.06
6 1.
069
1.07
2 1.
076
1.07
8
I2
1.05
2 1.
054
1.05
6 1.
059
1.06
1 14
1.
041
1.04
3 1.
045
1.04
7 1.
048
16
1.03
4 1.
035
1.03
7 1.
038
1.04
0 18
1.
028
1.02
9 1.
031
1.03
2 1.
033
20
1.02
4 1.
025
1.02
6 1.
027
1.02
8
24
1.01
8 1.
019
1.01
9 1.
020
1.02
1 30
1.
012
1.01
3 1.
013
1.01
4 1.
014
40
1.00
8 1.
008
1.00
8 1.
009
1.00
9 60
1.00
4 1.
004
1.00
4 1.
004
1.00
4 12
0 1.
001
1.00
1 1.
001
1.00
1 1.
001
x
1.O
oo
1.OOo
1.O
Oo
1.OOo
1.O
Oo
x;",
50.6
60
54.5
72
58.1
20
62.4
28
65.4
76
1.42
7 1.
292
1.21
7 1.
171
1.13
8 1.
115
1.09
7 1 .ow
1.07
3
1.05
7 1.
046
1.03
7 1.
03 1
1.02
7
1.02
0 1.
014
1.00
9 1.
004
1.00
1
1 .M
)o
54.0
902
r=1
3
r=l4
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.050
0.02
5 0.
010
0.W5
2.02
6 2.
095
1.78
0 1.
857
1.93
1 1.
458
1.30
9 1.
229
1.17
9
1.14
5 1.
121
1.10
2 1.
088
1.07
6
1.05
9 1.
048
1.03
9 1.
033
1.02
8
1.02
1 1.
015
1.00
9 1.
004
1 .00
1
1 .OOo
58.1
240
1.48
6 1.
326
1.24
0 1.
188
1.15
2 1.
126
1.10
6 1.
091
1.07
9
1.06
1 1.
049
1.04
1 1.
034
1.02
9
1.02
2 1.
015
1.00
9 1.
005
1.00
1
1 .Ooo
6 1.
7768
1.52
3 1.
346
1.25
4 1.
198
1.15
9 1.
132
1.11
1 1.
095
1.08
2
1.06
4 1.
052
I .04
2 1.
035
1.03
0
1.02
3 1.
016
1.01
0 1.
005
I .00
1
1 .Ooo
66.2
062
1.54
9 1.
361
1.26
4 1.
205
1.16
5 1.
136
1.1
15
1.09
9 1 .OM
1.
066
1.05
3 1.
044
1.03
6 1.
03 1
1.02
3 1.
016
1.01
0 1.
005
1.00
1
1 .Ooo
69.3
360
r=I5
r=
16
a
0.10
0 0.
050
0.05
0 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
I 1.
808
1.88
7 1.
964
2.06
1 2.
133
1.33
5 1.
995
2.095
2
1.44
9 3
1.30
9 4
1.23
2 5
1.183
6 1.
149
7 1.
124
8 1.
105
9 1.
091
10
1.07
9
12
1.06
2 14
1.
050
16
1.04
1 18
1.
035
20
1.03
0
24
1.02
2 30
1.
016
40
1.01
0 60
1.00
5 12
0 1.
001
00
1.OOo
x;, 57
.505
1.48
0 I .3
27
1.24
4 1.
192
1.15
6 1.
130
1.1 1
0 1.
095
1.08
3
1.06
5 I .0
52
1.04
3 I .0
36
1.03
1
I .023
1.
016
1.01
0 1.
005
1.00
1
I .Ooo
6 1.6
56
1.51
0 1.
344
1.25
6 I .z
oo
1.16
3 1.
135
1.11
5 I .099
I .08
6
1.06
7 1.
054
1.04
5 1.
037
1.03
2
1.02
4 1.
017
1.01
0 1.
005
1.00
1
1 .Ooo
65.4
10
1.54
7 1.
365
1.27
0 1.
211
1.17
1 1.
142
1.12
0 1.
103
1.09
0
I .07
0 I .0
56
I .04
7 1.
039
1.03
3
1.02
5 1.
017
1.01
1
I .00
5 1.
001
1 .Ooo
69.9
57
1.57
5 1.3
81
I .280
1.
218
1.17
7 1.
147
1.12
4 1.
107
1.093
1.07
2 1.
058
1.04
8 I .040
1.03
4
I .026
1.
018
1.01
I 1.
005
1.00
2
1 .OOo
73
.166
1.46
9 1.
325
I .245
1.1
95
1.15
9 1.1
33
1.1 1
4 1.
098
1.08
5
1.06
7 1.
054
1.04
5 1.
038
1.03
2
I .025
1.01
7 1.
01 I
1.00
5 1.
002
1 .Ooo
1.916
1.
501
1.34
4 I .2
58
1.20
4
1.16
7 1.
139
1.11
9 1.
102
I .089
1.07
0 1.
057
1.04
7 1.
039
1.03
4
1.02
6 1.
018
1.01
1 1.
006
1.00
2
I .Ooo
1.532
I .3
62
1.27
1 1.
213
1.17
4 1.
145
1.123
1.
106
1.09
2
1.01
3 1.0
59
1.049
1.
041
1.035
1.026
1.0
18
1.01
1
1.00
6 1.0
02
I .Ooo
1.57
1 1.
384
1.28
5 1.
224
1.18
2 1.
152
1.12
9 1.
111
1.09
7
1.07
6 1.
061
1.05
1 I .0
43
1.03
6
I .027
1.
019
1.01
2 I .0
06
1.00
2
1 .Ooo
2.1%
1.
599
I .400
I .2
%
1.23
2
1.18
8 1.
157
1.13
3 1.1
15
1 .ow
1.07
8 1.
063
I .05
2 I .044
1.03
7
1.02
8 1.
020
1.01
2 1.
006
1.00
2
1 .Ooo
60.9066
65.1
708
69.0
226
73.6
826
76.%
88
Tab
le 9
(C
ontin
ued)
-
m=
3 -
1x
17
r=
18
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1 1.
861
1.94
4 2.
025
2.12
7 2.
203
1.88
6 1.
971
2.05
3 2.
158
2.23
5 2
1.48
9 1.
522
1.55
4 1.
594
1.62
3 1.
508
1.54
2 1.
575
1.61
6 1.
646
3 1.
341
1.36
1 1.
379
1.40
2 1.
419
1.35
7 1.
377
1.3%
1.
420
1.43
7 4
1.25
9 1.
273
1.28
5 1.
300
1.31
2 1.
272
1.28
6 1.
299
1.31
5 1.
327
5 1.
206
1.21
6 1.2
25
1.23
7 1.
245
1.21
8 1.
228
1.23
8 1.
249
1.25
8
6 1.
169
1.17
7 1.
184
1.19
3 1.
200
1.17
9 1.
188
1.19
5 1.
204
1.21
1 7
1.14
2 1.
149
1.15
4 1.
162
1.16
7 1.
151
1.15
8 1.
164
1.17
1 1.
177
8 1.
122
1.12
7 1.
132
1.13
8 1.
142
1.12
9 1.
135
1.14
0 1.
146
1.15
1 9
1.10
5 1.
110
1.11
4 1.
119
1.12
3 1.
112
1.11
7 1.
121
1.12
7 1.
130
10
1.09
2 1.
096
1.10
0 1.
104
1.10
7 1.
099
1.10
3 1.
107
1.i
ll
1.11
4
12
1.07
3 1.
076
1.07
9 1.
082
1.08
4 1.
078
1.08
1 1.
084
1.08
7 1.
090
14
1.05
9 1.
061
1.06
4 1.
066
1.06
8 1.
064
1.06
6 16
1.
049
1.05
1 1.
053
1.05
5 1.
056
1.05
3 1.
055
18
1.04
1 1.
043
1.04
4 1.
046
1.04
7 1.
045
1.04
6 20
1.
035
1.03
7 1.
038
1.04
0 1.
041
1.03
8 1.
040
24
1.02
7 1.
028
1.02
9 1.
030
1.03
1 1.
029
1.03
0 30
1.
019
1.02
0 1.
020
1.02
1 1.
022
1.02
1 1.
021
.068
1.
071
.057
1.
059
.048
1.
050
.MI
1.04
3
.031
1.
032
.022
1.02
3
.073
.0
6 1
.05 1
.0
44
.033
.0
23
40
1.01
2 1.
012
1.01
3 1.
013
1.01
3 1.
013
1.01
3 1.
014
1.01
4 1.
015
60
1.00
6 1.
006
1.00
6 1.
006
1.00
7 1.
006
1.00
7 1.
007
1.00
7 1.
007
120
1.00
2 1.
002
1.00
2 1.
002
1.00
2 1.
002
1.00
2 1.
002
1.00
2 1.
002
00
1.Ooo
1.O
Oo
1.Ooo
1.O
Oo
1.Ooo
1.O
Oo
1.OOo
1.m
1.O
Oo
l.m
x:, 64
.295
68
.669
72
.616
77
.386
80
.747
67
.672
8 72
.153
2 76
.192
0 81
.068
8 84
.501
9
r=19
r =20
0.100
0.05
0 0.025
0.010
0.005
0.100
0.050
0.025
0.010
0.005
1 1.909
2 1.526
3 1.372
4 1.285
5 1.229
6 1.189
7 1.
160
8 1.137
9 1.119
10
1.10
5
12
1.084
14
1.068
16 1.057
18
1.048
20
1.041
24
1.032
30
1.022
40
1.014
60
1.007
120
1.002
M
1.OOo
xf,
71.040
1.9%
1.56 1
1.393
1.300
I .240
1.198
1.167
1.143
1.124
1.109
I .087
I .07 1
1.059
1.050
1.043
1.033
I .023
1.015
1.007
1.002
I .ooo
75.624
2.080
1.595
1.412
1.313
1.250
1.205
1.173
1.148
1.129
1.1 13
1.090
1.073
1.061
1.052
1.04
4
1.034
1.024
1.015
1.008
1.002
1 .Ooo
79.752
2.188
1.637
1.437
1.330
I .262
1.215
1.181
1.155
1.134
1.118
1.093
1.076
1.063
1.054
1.04
6
1.035
1 .O25
1.016
1.008
1.002
I .Ooo
84.733
2.261
1.668
1.454
1.341
1.271
1.222
1.186
1.159
1.138
1.121
1 .O%
1.078
1.065
1.055
1.047
1.036
1.025
1.016
1.008
1.002
I .Ooo
88.236
1.932
1.54
4 1.387
1.298
1.240
1.199
1.168
1.145
1.127
1.112
1.089
1.073
1.061
I .052
1.04
4
1.034
1.024
1.015
I .008
1.002
1 .Ooo
2.021
1.580
1 .a8
1.313
1.251
1.208
1.176
1.151
1.132
1.1 I6
I .092
I .075
1.063
1.053
1.04
6
1.035
I .025
1.016
1.008
1.002
I .Ooo
2.106
1.614
1.428
1.327
1.26 1
1.216
1.182
1.157
1.136
1.120
I .095
1.078
I .065
1.055
1.048
1.036
1.026
1.016
1.008
1.002
1 .O
oo
2.216
1.657
1.453
I .344
1.274
1.226
1.190
1.163
1.142
1.125
1.099
1.08 I
1.067
1.057
I .049
1.038
1.027
1.017
1.009
1.002
I .ooo
2.297
1.689
1.472
1.356
1.283
1.233
1.1%
1.168
1.146
1.128
1.102
1.083
1.069
1.059
1.05
0
1.039
1.027
1.017
1.009
1.003
1 .O
oo
74.3970
79.0819
83.2976
88.3794
91.9517
Tab
le9
(Can
linrr
ed)
m =
3 -
r =
21
r=2
2
a 0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.025
0.01
0 0.
005
1 1.
954
2.04
4 2.
131
2.24
3 2.
325
1.97
5 2.
067
2.15
6 2
1.56
1 3
1.40
1 4
1.31
0 5
1.25
0
6 1.
208
7 1.
177
8 1.
153
9 1.
133
10
1.11
8
12
1.09
4 14
1.
077
16
1.06
5 18
1.
055
20
1.04
8
24
1.03
6 30
1.
026
40
1.01
6 60
1.00
8 12
0 1.
002
30
1.OOo
xf,,
77.7
45
1.59
8 1.
423
1.32
5 1.
262
1.21
7 1.
184
1.15
9 1.
139
1.12
2
1.09
8 1.
080
1.06
7 1.
057
1.04
9
1.03
8 I .0
27
1.01
7 1.
009
1.00
2
i .OOo
82.5
29
1.63
3 1.
677
1.44
4 1.
470
1.34
0 1.
357
1.273
1.
286
1.22
6 1.
236
1.19
1 1.
200
1.16
5 1.
172
1.14
4 1.
150
1.12
7 1.
132
1.10
1 1.
105
1.08
3 1.
086
1.06
9 1.
072
1.05
9 1.
061
1.05
1 1.
053
1.03
9 1.
040
1.02
8 1.
029
1.01
8 1.
018
1.00
9 1.
009
1.00
3 1.
003
1.ooo
1.O
Oo
86.8
30
92.0
10
1.70
9 1.
488
1.37
0 1.
295
1.24
3 1.2
05
1.17
6 1.
154
1.13
5
1.10
8 1.
088
1.07
4 1.
063
1.05
4
1.04
1 1.
029
1.01
9 1.
009
1.00
3
1 .OOo
95.6
49
1.578
1.
415
1.32
2 1.
261
1.21
8 1.
185
1.16
0 1.
142
1.12
4
1.09
9 1.
082
1.06
9 1.
059
1.05
1
1.03
9 1.
028
1.01
8 1.
009
1.00
3
1 .Ooo
1.61
6 1.
651
1.43
8 1.
459
1.33
8 1.
353
1.27
3 1.
284
1.22
7 1.
236
1.19
3 1.
200
1.16
7 1.
173
1.14
7 1.
151
1.12
9 1.
133
1.10
3 1.
106
1.08
5 1.
087
1.07
1 1.
073
1.06
1 1.
063
1.05
2 1.
054
1.04
0 1.
041
1.02
9 1.
030
1.01
8 1.
019
1.00
9 1.
010
1.00
3 1.
003
1.Ooo
1.O
oo
2.26
9 2.
353
1.6%
1.
729
1.48
5 1.
514
1.37
1 1.
384
1.29
7 1.
307
1.24
6 1.
254
1.20
9 1.
213
1.18
0 1.
183
1.15
7 1.
161
1.13
9 1.
141
1.11
0 1.
115
1.09
1 1.
093
1.07
6 1.
078
1.06
5 1.
065
1.05
6 1.
057
1.04
3 1.
044
1.03
1 1.
031
1.02
0 1.
020
1.01
0 1.
010
1.00
3 1.
0oO
1.Ooo
1.O
oo
81.0
855
85.9
649
90.3
489
95.6
257
99.3
36
m =
4 -
r =
4
r =5
0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
I 1.
405
2 1.
178
3 1.
105
4 1.
071
5 1.
051
6 1.
039
7 1.
031
8 1.
025
9 1.
020
10
1.01
7
12
1.01
3 14
1.
010
16
1.00
8 18
1.
006
20
1.00
5
24
1.00
4 30
1.
002
40
1.00
1 60
1.
001
120
1.O
oo
DTJ
1.OOo
1.45
1 1.
194
1.11
4 1.
076
1.05
5
1.04
2 1.
033
1.02
7 I .
022
1.01
8
1.01
4 1.
010
I .00
8 1.
007
I .006
I .004
1.
003
1.00
2 1.
001
1 .Ooo
1 .Ooo
1 .49
4 I .2
09
1.12
2 I .0
8 1
1.05
8
I .044
1.03
5 1.
028
I .023
1.
019
1.01
4 1.0
1 I
1.00
9 1.
007
1.00
6
I .004
I .003
1.
002
1.00
1 1 .O
oo
I .Ooo
IS50
I .2
29
1.13
2 I .0
88
1.06
3
1.04
8 1.
037
1.03
0 I .0
25
1.02
1
1.01
5 1.
012
1.00
9 1.
008
1.00
6
I .004
I .0
03
I .002
1.
001
I .Ooo
I .Ooo
1.58
9 I .2
43
1.13
9 1.
092
1.06
6
1 .OM
1.03
9 1.
032
1.02
6 1.
022
1.01
6 1.
012
1.01
0 I .008
1.00
7
I .005
1.
003
1.00
2 1.
001
1 .Ooo
I .Ooo
~5
, 23.
5418
26
.296
2 28
.845
4 31
.999
9 34
.267
2
I .43
5 1.
199
1.12
1 1.
083
1.06
1
1.04
7 1.
037
1.03
0 I .
025
1.02
1
1.01
6 1.
01 2
1.01
0 1.
008
1.00
7
I .005
I .0
03
1.00
2 1 .
001
1 .Ooo
I .O
oo
1 .48
3 1.
216
1.13
0 1.
089
I .06
5
1.05
0 1.
040
1.03
2 1.
027
1.02
3
1.01
7 1.
013
I .ow
1.00
8 1.
007
1.00
5 I .
003
1.00
2 I .
001
1 .Ooo
I .OOo
1.53
0 1.
233
1.13
9 1.
094
1.06
9
1.05
3 I .
042
1.03
4 1.
028
1.02
4
1.01
8 1.
014
1.01
1 1.
009
1.00
7
I .005
1.
004
1.00
2 1.
001
1 .Ooo
1 .m
1.58
9 1.
253
1.15
0 1.
101
I .07
4
I ,05
6 1.
044
1.03
6 1.
030
I .02
5
1.01
9 1.
014
1.01
2 1.
009
1.00
8
I .006
I .00
4 1.
002
1.00
1 1 .O
Oo
1 .OOo
I .63
2 1.
269
1.15
8 1.
106
I .077
I .05
9 1.
046
1.03
8 I .
03 I
1.02
6
1.02
0 1.
015
1.01
2 1.
010
1.00
8
1.00
6 1.
004
I .00
2 1.
001
1 .OOo
I .O
Oo
28.4
120
3 1.
4104
34
.16%
37
.566
2 29
.996
8
m =
4 -
1 =
6 r
=7
0.02
5 0.
010
0.00
5 0.
100
0.050
0.02
5 0.
010
0.00
5
1 1.
466
2 1.
222
3 1.
138
4
1.0%
5
1.07
1
6 1.
055
7 1.
044
8 1.
036
9 1.
030
10
1.02
6
12
1.01
9 14
1.
015
16
1.01
2 18
1.
010
20
1.00
8
24
1.00
6 30
1.
004
40
1.00
2 60
1.
001
120
1.OOo
30
1.Ooo
1.51
7 1.
566
1.24
0 12
57
1.14
8 1.
157
1.10
2 1.
108
1.07
6 1.
080
1.05
9 1.
062
1.04
7 1.
049
1.03
8 1.
040
1.03
2 1.
034
1.02
7 1.
029
1.02
0 1.
021
1.01
6 1.
017
1.01
3 1.
013
1.01
0 1.
011
1.00
9 1.
009
1.00
6 1.
007
1.00
4 1.
004
1.00
2 1.
003
1.00
1 1.
001
1.O
oo
1.OOo
1.OOo
1.O
oo
1.62
8 1.
279
1.16
8 1.
1 15
1 .
085
1.06
6 1.
052
1.04
3 1.
036
1.03
0
1.02
3 1.
018
1.01
4 1.
012
1.01
0
I .007
I .005
I .003
1.00
1 1 .o
oo 1 .O
oo
1.67
4 1.
295
1.17
7 1.
121
1.08
9
1.06
8 1.
055
1.04
5 1.
037
I .032
1.02
4 1.
018
1.01
5 1.
012
1.01
0
1 .00
7 1.
005
1.00
3 1.
001
1 .Ooo
1 .Ooo
x$,
33.1
963
36.4
151
39.3
641
42.9
798
45.5
585
1.49
7 1.
550
1.24
4 1.
263
1.15
5 1.
165
1.10
9 1.
116
1.08
2 1.
087
1.06
4 1.
068
1.05
2 1.
055
1.04
3 1.
045
1.03
6 1.
038
1.03
1 1.
032
1.02
3 1.
024
1.01
8 1.
019
1.01
5 1.
015
1.01
2 1.
013
1.01
0 1.
011
1.00
7 1.
008
1.00
5 1.
005
1.00
3 1.
003
1.00
1 1.
001
1.OOo
1.O
Oo
1.m
1.O
oo
1.60
1 1.
281
1.17
5 1.
122
1.09
2
1.07
1
1.05
7 1.
047
1.04
0 1.
034
1.02
6 1.
020
1.01
6 1.
013
1.01
1
1.00
8 1.0
05
1.00
3 1.
002
1 .Ooo
1 .ooo
1.66
7 1.
305
1.18
8 1.
130
1.09
7
1.07
6 1.
061
1.05
0 1.
042
1.03
6
I .027
I .0
2 1
1.01
7 1.
014
1.01
2
1.00
8 1.0
06
1.00
3 1.
002
1 .O
oo
1 .Ooo
1.71
5 1.
322
1.19
7 1.
136
1.10
1
1.07
9 1.
063
1.05
2 1.
044
1.03
7
I .02
8 1.
022
1.01
7 1.
014
1.01
2
1.00
9 I .0
06
1.00
4 I .0
02
1 .Ooo
I .O
oo
37.9
159
41.3
372
44.4
607
18.2
782
50.9
933
r =8
r=9
a
0.10
0 1.
050
1.025
0.01
0 0.
005
0.100
0.050
0.025
0.010
0.00
5
1.583
1.636
1 2 3 4 5 6 7 8 9 10 12
14
16 18
20
24
30
40
60
120 a0
I .528-
1.266
1.172
1.123
1.093
1.074
I .060
1.050
1.04
2 1.036
1.027
1.02 1
1.107
I .Ol4
1.012
1.009
1.006
I .003
1.002
I .Ooo
1 .O
oo
1.286
1.183
1.130
I .099
1.078
1.063
1.052
1.04
4 1.038
1.029
I .023
1.10
8 1.
015
1.013
1.009
1.00
6 1.
004
1.002
1 .Ooo
1 .Ooo
1.305
1.193
1.137
1.103
1.08 1
I .066
1.055
1.04
6 I .039
1.030
1.023
1.109
1.016
1.013
1.010
1.007
1.00
4 I .002
1 .Ooo
1 .Ooo
1 .-704
1.330
1.207
1.146
1.109
1.086
1.070
1.058
1.048
1.041
1.03 1
I .025
1.
020
1.016
1.01
4
1.010
1.007
1.00
4 I .002
1.001
1 .Ooo
1.754
~ 1.557
1.348
1.216
1.152
1.1 I4
1.089
1.072
1.06
0 1.050
1.043
.033
.026
.02 1
.017
.014
.010
1.007
1.00
4 I .002
1.001
I .Ooo
x:,
42.5847
46.1943
49.4804
53.4858
56.3281
1.288
1.189
1.137
1. I05
1.083
1.068
1.05
7 1.048
1.041
I .032
1.025
1.02
0 1.017
1.014
1.010
1.007
1.00
4 I .002
1.001
I .Ooo
1.6 I4
1.309
1.201
1.144
1.1
10
I .08
8 I .07 I
1.06
0 1.
050
.043
1.033
1.026
1.02 1
1.018
1.01
5
1.01 1
1.007
1.00
4 I .0
02
1.001
1 .OO0
I .669
1.740
1.792
1.329
1.212
1.15
2 1.115
1.09 1
1.075
1.062
1.053
1.045
1.034
1.027
I .022
1.018
1.015
1.011
1.008
1.005
1.002
1.001
I .Ooo
1.355
1.226
1.161
1.122
I .O%
1.078
1.065
1.055
I .047
1.036
1.029
I .023
1.019
1.016
1.012
1.008
I .005
I .002
1.001
1 .Ooo
1.373
1.236
1.167
1.12
7
1.10
0 1 .o
s 1
1.068
1.057
1.049
1.037
1.02
9 1.
024
1.02
0 1.017
1.012
1.008
1.005
I .002
1.001
1 .Ooo
47.2122
50.9985
54.4373
58.6192
61.5812
Tab
le9
(Con
linue
d)
m =
4
r=lO
r=
ll
n
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
1 20 m
Xr”
, 2
1.58
5 1.
309
1.20
6 1.
150
1.1 1
6
1.09
3 1.
076
1.06
4 1 .OM
1.03
7
1.03
6 1.
029
I .02
3 1.
019
1.01
6
1.01
2 I .
008
1.00
5 1.
002
1.00
1
I .Ooo
5 1
.805
0
~
I .644
1.33
1 1.
218
1.15
9 1.
122
1.09
7 1.
080
1.06
7 1.
057
1.04
9
1.03
8 1.
030
1.02
4 1.
020
1.01
7
1.01
3 1.
009
1.00
5 1.
003
1.00
1
1 .ooo
-55.
7585
1.70
1 1.
352
1.23
0 1.
166
1.12
8
1.10
2 I .
083
1.07
0 1.
059
1.05
1
1.03
9 1.
03 1
1.
025
1.02
1 1.
018
1.01
3 1.
009
1.00
5 1.
003
1.00
1
1 .Ooo
59.3
41 7
1.77
4 1.
379
1.24
4 1.
176
1.13
4
1.10
7 1.
088
1.07
3 1.
062
1.05
4
1.04
1 1.
033
I .026
1.
022
1.01
9
1.01
4 1.
009
1.00
6 1.
003
1.00
1
1 .Ooo
63
.690
7
1.82
8 1.
398
1.25
5 1.
183
1.13
9
1.11
1 1.
090
1.07
6 1.
064
1.05
5
1.04
2 1.
034
1.02
7 1.
023
1.01
9
1.01
4 1.
010
1.00
6 1.
003
1.00
1
1 .Ooo
66.7
659
-
1.33
0 1.
222
1.16
4 1.
127
1.10
3 1 .OM
1.
071
1.06
1 1.
053
1.04
1 1.
033
1.02
7 1.
022
1.01
9
1.01
4 1.
010
1.00
6 1.
003
1.00
1
1 .O
oo
56.3
69
-
1.35
2 1.
235
1.17
3 1.
134
1.10
7 1.
089
1.07
5 1.
064
1.05
5
1.04
3 1.
034
1.02
8 1.
023
1.02
0
1.01
5 1.
010
1.00
6 1.
003
1.00
1
1 .O
oo
60.4
8 1
-
1.37
4 1.
247
1.18
1 1.
140
1.1 1
2 1.
092
1.07
7 1.
066
I .057
1.04
4 1.
035
1.02
9 1.
024
1.02
0
1.01
5 1.
010
1.00
6 1.
003
1.00
1
1 .Ooo
64.2
01
-
1.40
2 1.
262
1.19
1 1.
147
1.11
8 1.
097
1.08
1 1.
069
1.06
0
1 .M
1.
037
1.03
0 1.
025
1.02
1
1.01
6 1.
01 1
1.00
7 1.
003
1.00
1
1 .Ooo
68
.710
-
1.42
2 1.
214
1.19
8 1.
152
1.12
2 1.
100
1 .OM
1.07
1 1.
062
1.04
7 1.
038
1.03
1 1.
026
1.02
2
1.01
6 1.
01 1
1.
007
1.00
3 1.
001
1 .Ooo
71.8
93
r=1
2
r=1
3
u
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
I 20 oc
1.63
8 1.
350
1.23
8 1.
177
1.13
9
1.1
12
1.09
3 1.
079
1.06
8 1.
059
1.04
6 1.
037
1.03
0 1.
025
I .02
1
1.01
6 1.
01 I
1.00
7 1.
003
I .00
1
I .Ooo
1.70
0 1.
373
1.25
2 1.
186
1.14
5
1.11
8 I .
097
1.08
2 1.
070
1.06
1
1.04
7 1.
038
1.03
1 I .
026
1.02
2
1.01
7 1.
01 1
1.00
7 1.
003
I .001
1 .Ooo
1.76
0 I .
3%
1.26
4 .I
95
.I52
.I22
,1
01
.085
.0
73
.063
1.83
8 I .4
24
1.28
0 1.
205
1.15
9
1.12
8 1.
106
I .089
1.
076
1.06
6
1.89
5 1.
446
I .292
1.
213
1.16
5
1.13
2 1.
109
I .092
I .
079
1.06
8
I .049
1.
039
I .032
1.
027
1.02
3
1.01
7 1.
012
I .007
1.
004
1 .00
1
1 .Ooo
.05 1
.04 1
,0
33
.028
.0
24
.018
1.
012
1.00
8 1.
004
1.00
1
I .Ooo
,053
,0
42
.034
.0
29
.024
.018
1.
013
1.00
8 1.
004
1.00
1
I .OOo
xf
,, 60
.906
6 65
.170
8 69
.022
6 73
.682
6 76
.%88
-
1.36
9 1.
254
1.19
0 1.
150
1.12
2 I.
102
1.08
6 1.
074
I .065
,050
.0
4 1
.033
,0
28
,024
,018
1.
012
1.00
8 I .0
04
1.00
1
1 .OOo
-
1.39
3 1.
268
I .20
0 1.
157
1.12
7 1.
106
1.09
0 1.
077
1.06
7
I .05
2 I .
042
1.03
5 1.
029
1.02
5
1.01
9 1.
01 3
1.00
8 1.
004
1 .00
1
1 .Ooo
-
1.41
7 1.
281
1.20
9 1.
163
1.13
2 1.
110
I .09
3 1.
080
1.07
0
1.05
4 1.
044
1.03
6 1.
030
1.02
6
1.01
9 1.
013
1.00
8 1.
004
1 .001
1 .Ooo
-
1.44
6 1.
298
I .22
0 1.
171
1.13
9 1.
115
I .09
7 1.
083
1.07
3
1.05
6 I .
045
1.03
7 I .0
3 I
I .02
7
1.02
0 1.
014
1.00
8 1.
004
I .00
1
1 .Ooo
65.4
22
69.8
32
73.8
10
78.6
16
-
I .468
1.31
0 I .
228
1.17
7
1.14
3 1.
118
1.10
0 1.
086
1.07
5
1.05
8 1.
047
I .03
8 1.
032
1.02
7
1.02
0 1.
014
1.00
9 1.
004
1.00
1
1 .Ooo
82.0
01
f
0
Tab
le9
(Con
tin
ued
) -
~~
~
m =
4 -
r=1
4
r=1
5
Q
0.10
0 0.050
0.02
5 0.
010
0.00
5 0.
100
0.050
0.02
5 0.
010
0.00
5
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
120 co
I .686
I .3
88
1.26
9 1.
203
1.16
1
1.13
1 1.
110
1.09
4 1.
08 1
1.0
7 I
1 .OH
1.
045
1.03
7 I .0
3 1
1.02
6
1.02
0 1.
014
I .009
1.
004
1.00
1
1 .Ooo
1.75
1 1.
413
1.28
4 1.
213
1.16
8
1.13
7 1.
115
1.09
7 1.
084
1.07
3
1.05
8 I .
046
I .03
8 1.
032
1.02
7
1.02
1 1.
014
1.00
9 1.
004
1.00
1
1 .ooo
1.81
4 1.
436
1.29
7 1.
222
1.17
5
1.14
2 1.
119
1.10
1 1.
087
1.07
6
1.05
9 1.
048
1.03
9 1.
033
1.02
8
1.02
1 1.
015
1.00
9 1.
005
1.00
1
1 .Ooo
1.8%
1.
467
1.31
4 1.
234
1.18
3
1.14
9 1.
124
1.10
5 1.
09 1
1.07
9
1.06
2 1.
050
1.04
1 1.
034
I .02
9
1.02
2 1.
015
1.00
9 I .0
05
1.00
1
1 .ooo
1.95
6 I .
489
1.32
7 1.
242
1.18
9
1.15
4 1.
128
1.10
9 1.
093
1.08
1
1.06
4 1.
05 1
1.
042
1.03
5 1.
030
I .02
3 1.
016
1.01
0 1.
005
1.00
1
1 .ooo
x?,
69.9
185
74.4
683
78.5
671
83.5
134
86.9
937
-
1.40
6 1.
284
1.21
6 1.
172
1.14
1 1.1
18
1.10
1 1.
087
1.07
7
1 .06
0 1.
049
1 .oQo
1.03
4 1.
029
1.02
2 1.
015
I .01
0 1.
005
1.00
1
I .ooo
74.3
97
-
1.43
2 1.
299
1.22
6 1.
179
1.14
7 1.
123
1.10
5 1.0
9 1
1.08
0
1.06
3 1.0
5 1
I .042
1.
035
1.03
0
1.02
3 1.
016
1.01
0 1.
005
1.00
1
1 .Ooo
79.0
82
-
1.45
6 1.
313
1.23
6 1.
187
1.15
3 1.
128
1.10
9 1 .o
w
1.08
2
1.06
5 1.
052
1.04
3 I .
036
1.03
1
1.02
3 1.
016
1.01
0 1.
005
1.00
1
1 .ooo
83.2
98
-
1.48
8 1.
331
1.24
8 1.
195
1.15
9 1.
133
1.1 1
3 1.
098
1.08
5
1.06
7 1.
054
1.04
5 1.
038
1.03
2
1.02
4 1.
017
1.01
1 1.
005
1.00
1
1 .Ooo
83.3
79
-
1.51
1 1.
344
1.25
6 1.
202
1.16
4 1.
137
1.1 I
6
1.10
1 1.
088
1.06
9 1.
056
1.04
6 1.
039
1.03
3
1.02
5 1.
017
1.01
1
1.00
5 1.
001
1 .ooo
91.9
52
r=16
r
= I
7 0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 1.
731
2 1.
423
3 1.
299
4 1.
223
5 1.
182
6 1.
150
7 1.
127
8 1.
108
9 1.
094
10
1.08
3
12
1.06
5 14
1.
053
16
1.04
4 18
1.
037
20
1.03
2
24
1.02
4 30
1.
017
40
1.01
1 60
1.00
5 12
0 1.
001
03
1.OOo
1.79
9 I .
450
1.31
4 1.
239
1.19
0
1.15
7 1.
132
1.1
I3
1.09
8 I .0
86
I .06
8 I .
055
1.04
5 1.
038
1.03
3
1.02
5 1.
018
1.01
1 I .
00s
1.00
2
1 .Ooo
I .86
4 1.
475
1.32
9 1.
249
1.19
8
1.16
3 1.
136
1.11
7 1.
101
1.08
9
1.07
0 1.
056
I .047
1.
040
1.03
4
1.02
6 1.
018
1.01
1 I .006
1.00
2
I .Ooo
I .949
I S
O7
I .347
1.
26 1
I .207
1.16
9 1.
142
1.12
1 1.
105
1.09
2
1.07
3 1.
058
I .04
9 1.
041
1.03
5
1.02
7 1.
019
1.01
2 1.
006
I .002
1 .Ooo
2.0
12
1.53
1 1.
360
1.27
0 1.
213
1.17
4 1.
146
1.12
5 1.
108
1.09
4
,074
.0
6Q
,050
,0
42
,036
,027
1.
019
1.01
2 I .006
1.00
2
1 .Ooo
2
X,, 78
.859
7 83
.675
3 88
.004
0 93
.216
8 %
.878
1
-
-
-
-
-
1.44
0 1.
468
1.49
4 1.
527
1.55
1 1.
313
1.32
9 1.
344
1.36
3 1.
377
1.24
0 1.
252
1.26
2 1.
275
1.28
4 1.
193
1.20
1 1.
209
1.21
8 1.
225
1.16
0 1.
166
1.17
2 1.
180
1.18
5 1.
135
1.14
0 1.
145
1.15
1 1.
155
1.11
6 1.
120
1.12
4 1.
129
1.13
3 1.
101
1.10
5 1.
108
1.11
2 1.
115
1.08
9
1.07
0 1.
057
1.04
8 I .
040
1.03
5
I .026
,092
1.
095
1.09
8 1.
101
,073
1.
075
1.07
8 1.
080
,059
1.
061
1.06
3 1.
065
a9
1.
051
1.05
3 1.
054
,042
1.
043
1.04
5 1.
046
.036
1.
037
1.03
8 1.
039
.027
1.
028
1.02
9 1.
030
1.01
9 1.
019
1.02
0 1.
020
1.02
1 1.
012
1.01
2 1.
012
1.01
3 1.
013
1.00
6 1.
006
1.00
6 1.
006
1.00
7 1.
002
1.00
2 1.
002
1.00
2 1.
002
1.O
oo
1.OOo
1.O
Oo
1.00
0 1.O
Oo
83.3
08
88.2
50
92.6
89
98.0
28
101.
776
Tab
le 9
(C
onfi
nw
d)
m =
4 -
r=1
8
r=1
9
0.050
0.02
5 0.
010
0.005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 1.
7%
2 1.
457
3 1.
327
4 1.
252
5 1.
203
6 1.
169
7 1.
143
8 1.
123
9 1.
107
10
1.09
5
12
1.07
5 14
1.
061
16
1,05
1 18
1.
044
20
1.03
7
24
1.02
9 30
1.
020
40
1.01
3 60
1.
006
120
1.00
2
ffi
1.O
Oo
xs,
87.7
431
~~ 1.84
3 1.
485
I .343
1.
264
1.21
2
1.17
6 1.
149
1.12
8 1.
1 11
1.09
8
1.07
8 I .
063
1.05
3 1.
045
1.03
9
1.03
0 1.
02 1
1.01
3 1.
007
1.00
2
1 .Ooo
92.8
083
1.91
1 1.
511
1.35
9 1.
274
1.22
0
1.18
2 1.
154
1.13
2 1.
1 15
1.
101
1.08
0 1.
065
1 .OM
1.04
6 1.
040
! .03
0 1.
022
1.01
4 1.
007
1.00
2
1 .ooo
97.3
53 1
! .999
1.54
5 1.
378
1.28
7 1.
230
1.18
9 1.
160
1.13
7 1.
1 I9
1.
105
1.08
3 1.
068
1.05
6 I .
048
1.04
1
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .Ooo
I 0
2.8 1
6
1.06
5 1.
570
1.39
2 1.
297
1.23
7
1.19
5 1.
164
1.14
1 1.
122
1.10
8
1.08
5 1.
069
I .05
8 1.
049
1.04
2
1.03
2 1.
023
1.01
4 1.
007
I .00
2
1 .Ooo
106.
648
L
1.47
3 I .
340
1.26
4 1.
214
1.17
8 1.
151
1.13
0 1.
114
1.10
1
1.08
0 1.
066
1.05
5 1.
047
1.04
0
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .Ooo
92
.166
-
-
-
-
1.50
2 1.
529
1.56
3 1.
588
1.35
7 1.
373
1.39
3 1.
408
1.27
6 1.
287
1.30
0 1.
310
1.22
3 1.
231
1.24
1 1.
248
1.18
5 1.
191
1.19
9 1.
205
1.15
7 1.
162
1.16
9 1.
173
1.13
5 1.
140
1.14
5 1.
149
1.11
8 1.
122
1.12
6 1.
130
1.10
4 1.
107
1.11
1 1.
114
1.08
3 1.
086
1.08
9 1.
091
1.06
8 1.
070
1.07
3 1.
074
1.05
7 1.
059
1.06
1 1.
062
1.04
8 1.
050
1.05
1 1.
053
1.04
2 1.
043
1.04
4 1.
045
1.03
2 1.
033
1.03
4 1.
035
1.02
3 1.
023
1.02
4 1.
025
1.01
4 1.
015
1.01
5 1.
015
1.00
7 1.
007
1.00
8 1.
008
1.00
2 1.
002
1.00
2 1.
002
1.OOo
1.
ooo
1.Ooo
1.
Ooo
97.3
51
101.
999
107.
583
11 1.
495
r =20
r=
21
Q
0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 1.
812
2 1.
488
3 1.
353
4 1.
275
5 1.
224
6 1.
187
7 1.
159
8 1.
138
9 1.
121
10
1.10
7
12
1.08
6 14
1.
070
16
1.05
9 18
1.
050
20
1.04
3
24
1.03
3 30
1.
024
40
1.01
5 60
1.
008
120
1.00
2
oc
1.OOo
1.88
4 1.
518
1.37
1 1.
288
1.23
3
1.19
4 1.
165
1.14
3 1.
125
1.1
10
,088
,072
.0
6 I
,052
.0
45
.034
1.
024
1.01
6 1.
008
1.00
2
1 .OOo
1.95
4 1.
545
1.38
7 1.
299
1.24
1
1.20
1 1.
170
1.14
7 1.
129
1.11
4
1.09
1 I .0
74
1.06
2 I .053
1.04
6
1.03
5 1.
025
1.01
6 1.
008
1.00
2
1 .OOo
2.04
5 1.
580
I .408
1.31
3 1.
252
I .208
1.
177
1.15
3 1.
133
1.1
I8
1.09
4 1.
077
I .06
4 1.
055
I .047
I .03
6 1.
026
1.01
6 1.
008
I .002
1 .ooo
2.11
3 1.
606
1.42
2 1.
323
I .259
1.21
5 1.
182
1.15
7 1.
137
1.12
1
1 .O%
1.07
8 1.
066
1.05
6 1.
048
1.03
7 1.
026
1.01
7 1.
008
1.00
2
1 .Ooo
x:m
96
.578
2 10
1.87
9 10
6.62
9 11
2.32
9 11
6.32
1
-
-
-
-
-
1.50
4 1.
533
1.56
2 1.
598
1.62
4 1.
367
1.38
4 1.
401
1.42
2 1.
437
1.28
7 1.
299
1.31
1 1.
325
1.33
5 1.
234
1.24
3 1.
252
1.26
2 1.
270
1.1%
1.
203
1.21
0 1.
218
1.22
4 1.
167
1.17
3 1.
179
1.18
6 1.
190
1.14
5 1.
150
1.15
5 1.
160
1.16
4 1.
127
1.13
2 1.
136
1.14
0 1.
144
1.11
3 1.
116
1.12
0 1.
124
1.12
7
1.09
1 1.
094
1.09
6 1.
075
1.07
7 1.
079
1.06
3 1.
065
1.06
6 1.
054
1.05
5 1.
057
1.04
6 1.
048
1.04
9
1.03
6 1.
037
1.03
8
.099
1.
102
,082
1.
084
.069
1.
070
.059
1.
060
.051
1.
052
.039
1.
040
1.02
5 1.
026
1.02
7 1.
028
1.02
8 1.
016
1.01
7 1.
017
1.01
8 1.
018
1.00
8 1.
008
1.00
9 1.
009
1.00
9 1.
002
1.00
2 1.
003
1.00
3 1.
003
1.OOo
1.O
Oo
1.m
1.O
Oo
1.OOo
100.
980
106.
395
I1 1.
242
117.
057
121.
126
a’
W
m =
4 -
r=2
2
a
0.10
0 0.050
0.02
5 0.
010
0.00
5
1 1.
848
1.92
2 2
1.51
8 3
1.37
9 4
1.29
8 5
1.24
3
6 1.
204
7 1.
175
8 1.
152
9 1.
134
10
1.11
9
12
1.09
5 14
1.
079
16
1.06
6 18
1.
057
20
1.04
9
24
1.03
8 30
1.
027
40
1.01
7 60
1.
009
120
1.00
3
CQ
1.Ooo
xt,
105.
372
I .54
9 1.
397
1.31
0 1.
253
1.21
2 1.
181
1.15
7 1.
138
1.12
3
1.09
8 1.
08 1
1.06
8 1.
058
1.05
1
1.03
9 1.
028
1.01
8 1.
009
1.00
3
1 .Ooo
10.8
98
1.99
4 I .5
77
1.41
4 I .3
22
1.26
2
1.21
9 1.
187
1.16
2 1.
142
1.12
6
1.10
1 1.
083
1.07
0 1.
060
I .052
1.04
0 1.
029
1.01
8 I .009
1.00
3
1 .Ooo
115.
841
2.08
8 2.
158
1.61
4 1.
436
1.33
7 1.
273
1.22
8 1.
194
1.16
8 1.
147
1.13
0
1.10
4 1.
086
1.07
2 I .
062
1.05
3
1.04
1 1.030
1.01
9 1.
010
1.00
3
1 .O
oo
12 1.
767
1.64
1 1.
45 1
1.
347
1.28
1
1.23
4 1.
199
1.17
2 1.
151
1.13
4
1.10
7 1.
088
1.07
4 1.
063
I .05
5
1.04
2 1.
030
1.01
9 1.
010
1.00
3
1 .Ooo
25.9
13
m =
5 -
r=5
r=6
0.050
0.025
0.010
0.005
0.100
0.050
0.025
0.010
0.005
I 1.448
2 1.212
3 1.132
4 1.092
5 1.
068
6
1.053
7 (1.042
8 1.035
9 1.029
10
1.025
12
1.018
14
1.014
16
1.011
18
1.009
20
1.008
24
1.00
6 30
1.004
40
1.002
60
1.00
1 120
1.OOo
33
1.OOo
xs,
34.3816
1.4%
I .230
1.141
1.098
1.072
1.056
I .045
1.037
1.03 1
I .026
1.020
1.01
5 1.012
1.010
1.008
1.00
6 I .0
04
1.002
1.001
1 .Ooo
I .O
oo
37.6525
I .544
I .246
1.150
1.103
1.076
1.059
I .047
I .039
1.032
I .027
1.020
1.016
1.013
1.010
1.009
I .006
I .004
1.002
1.00
1 I .O
oo
1 .Ooo
40.6
465
1.604
1.267
1.161
1.1 10
1.
081
1.063
I .OM
1.041
1.034
I .029
1.022
1.01
7 1.013
1.01 I
1.009
I .007
1.00
4 1.003
1.001
1 .Ooo
1 .O
oo
44.3141
1.64
9 I .283
1.169
1.1 I6
1.085
1.06
5 1.052
1.043
1.035
1.03
0
I .022
1.017
1.014
1.01 1
1.00
9
I .007
I .oo5
1.003
I .00
1 I .O
oo
1 .OOo
46.9279
I .465
1.22
8 1.
144
1.102
1.077
1.060
1.048
I .040
I .034
1.029
1.022
1.017
I .014
1.01 1
1.009
1.007
I .005
1.003
1.001
1 .Ooo
1 .O
oo
40.2560
1.514
I .245
1.15
4 1.
108
1.08 1
I .063
I .05 I
1.042
1.035
1.030
1.023
1.014
1.012
1.010
1.007
1.005
1.003
1.001
1 .Ooo
I .O
oo
43.7730
1.018
1.563
1.262
1.163
1.1 14
1.08
5
I .066
1.053
I .044
1.037
1.03 1
1.024
1.019
1.01
5 1.012
1.010
1.007
I .005
I .003
1.001
1 .OOo
1 .OOo
46.9792
1.625
I .2M
1.175
1.121
1 .om
1.070
1.056
1.04
6 I .039
1.033
I .025
1.019
1.016
1.013
1.01 1
1.00
8 1.
005
I .003
1.001
1 .Ooo
I .O
oo
50.8922
1.67 1
1.300
1.183
1.127
1.09
4
I .073
I .059
1.048
1.04
0 1.034
1.026
I .020
1.01
6 1.013
1.01 1
I .008
1.005
1.003
I .002
I .Ooo
1 .Ooo
53.6720
Tab
le 9
( C
on
tin
Ud
)
m=
5 -
r=7
r=8
0.05
0 0.025
0.010
0.005
0.100
0.05
0 0.025
0.010
0.005
1 1.484
2 1.244
3 1.158
4 1.113
5 1.086
6 1.068
7 1.055
8 1.
046
9 1.038
10
1.033
12
1.025
14
1.020
16
1.016
18
1.013
20
1.011
24
1.008
30
1.005
40
1.003
60
1.002
120
1.OOo
03
1.Ooo
xf,
16.0588
1.535
1.262
1.168
1.119
1.090
1.07 1
1.058
1.048
1.040
1.035
1.026
1.02 1
1.017
1.014
1.012
1.008
1.006
1.003
1.002
I .Ooo
1 .Ooo
49.8019
1.584
1.280
1.177
1.125
1.095
1.074
1.060
1.05
0 1 .a
42
1.036
1.027
1.02 1
1.017
1.014
1.012
1.009
1.006
i .004
1.002
I .Ooo
1 .Ooo
53.2033
1.648
1.302
1.189
1.133
1.100
1.078
1.063
1.052
1.044
1.038
1.029
1.022
1.018
1.015
1.013
1.009
1-0
6
1.004
1.002
1 .Ooo
1 .O
oo
57.3421
1.695
1.319
1.198
1.139
1.104
1.081
1.066
1.05
4 1 -046
1.039
1.030
1.023
1.019
1.015
1.013
1.010
I .006
1.00
4 1.002
1 .Ooo
1 .O
oo
60.2748
1 SO
5 1.261
1.171
1.124
1.095
1.076
1.062
1.052
1.044
1.038
1.029
I .023
1.018
1.015
1.013
1.009
1.006
1.004
1.002
1.001
1 .Ooo
5 1 .SO50
1.556
1.280
1.182
1.131
1.100
1.079
1.065
1.054
1.046
1.039
1.030
1.024
1.019
1.016
1.013
1.010
1.007
1 .w
1.002
1.001
1 .Ooo
55.7585
i .007
1.298
1.192
1.137
1.105
1.083
1.068
1.056
1 .a48
1.041
1.03 1
I .025
1.020
1.017
1.014
1.010
1.007
1.00
4 I .002
1.001
1 .Ooo
59.34 17
1.672
1.321
1.204
1.145
1.110
1.087
1.071
1.059
1.050
I .043
1.033
1.026
1.021
1.017
1.01
5
1.01 1
1.007
1.004
1.002
1.001
1 .Ooo
63.6907
1.721
1.338
1.213
1.151
1.1 14
1.09
0 1.073
1.061
I .052
1.044
1.034
1.027
1.022
1.018
1.015
1.01 1
1.008
1.005
1.002
1.001
1 .Ooo
66.7659
r=9
r=
lO
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
I 1.
526
2 1.
278
3 1.
185
4 1.
136
5 1.
105
6 1.
084
7 1.
069
8 1.
058
9 1.
049
10
1.04
3
12
1.03
3 14
1.
026
16
1.02
1 18
1.
018
20
1.01
5
24
1.01
1 30
1.
008
40
1.00
5 60
1.00
2 I2
0 1.
001
w
1.OOo
I .578
I .2
98
1.1%
,1
43
.I 10
,088
.0
72
.060
.05 1
.044
1.03
4 I .0
27
1.02
2 1.
018
1.01
5
1.01
1 1.
008
1.00
5 1.
002
1.00
1
1 .O
oo
1.63
0 1.
316
1.20
6 1.
150
1.11
5
1.09
2 I .0
75
1.06
3 1.
053
I .046
1.03
5 I .
028
1.02
3 1.
109
1.01
6
1.01
2 1.
008
1 .00
5 I .
002
1.00
1
I .Ooo
1.69
7 1.
340
1.21
9 1.
158
1.12
1
1 .O%
1.07
9 1.
066
1.05
6 1.
048
1.03
7 1.
029
I .02
4 1.
020
1.01
7
1.01
2 I .
008
1.00
5 1 .0
02
1.00
1
I .OOo
1.74
6 1.
358
I .22
9 1.
164
1.12
5
1 .ow
1.
08 1
1.06
8 1.
058
1.05
0
I .03
8 1.
030
1.02
4 1.
020
1.01
7
1.01
3 1.
009
1 .00
5 1.
003
1.00
1
1 .O
oo
x:,,,
57.5
053
61.6
562
65.4
102
69.9
568
73.1
661
1.54
7 I .2
95
1.19
9 1.
147
1.11
5
1.09
2 1.
076
I .064
1.05
5 1.
048
1.03
7 1.
029
1.02
4 1.
020
1.01
7
1.01
3 1.
009
1.00
5 1.
003
1.00
1
I .Ooo
1.60
0 1.
315
1.21
1 15
5 ,1
20
,097
,0
80
,067
.0
57
'050
1.03
8 I .0
3 1
1.02
5 1.
021
1.01
8
1.01
3 1.
009
I .005
1.
003
1.00
1
I .Ooo
I .653
1.
334
I .22
1 1.
162
1.12
5
1.10
1 I .
083
I .070
1.
059
1.05
1
1.04
0 1.
032
1.02
6 1.
022
1.01
8
1.01
4 1.
009
1.00
6 1.
003
1.00
1
1 .Ooo
1.72
1 I ,
359
1.23
5 1.
171
1.131
1.105
I .087
1.
073
1.06
2 1.
054
1.04
1 1.
033
1.02
7 1.
022
1.01
9
1.01
4 1.
010
1.00
6 1.003
1 .#2
I .Ooo
I .77
2 1.
377
1.24
4 1.
177
1.13
6
1.10
9 I .
089
1.07
5 1.
064
I .05
5
1.04
3 1.
034
1.02
8 1.
023
1.01
9
1.01
4 1.
010
I .006
1.00
3 I .0
02
I .O
Oo
63.1
671
67.5
048
71.4
202
76.1
539
79.4
900
~ ~~
~~
m=
5 -
r=ll
?=I2
n
0.100
0.050
0.025
0.01
0 0.005
0.100
0.050
0.025
0.010
0.005
1-
2
1.312
3 1.213
4 1.159
5 1.125
6 1.101
7 1.084
8 1.071
9 1.061
10
1.053
12
1.041
14
1.033
16
1.027
18
1.023
20
1.019
24
1.014
30
1.01
0 40
1.00
6 60
1.003
I20
1.001
m
1.OOO
xf,
68.796
-
1.333
1.225
1.167
1.130
1.105
1.087
1.074
1.063
1.055
1.043
1 .OM
1.028
1.023
1.02
0
1.015
1.010
1.006
1.003
1.001
1 .ooo
73.3 1
1
I
1.352
1.236
1.174
1.136
1.110
1.09 1
1.077
1.06
6 1.057
I .044
1.035
1.029
I .024
I .021
1.01
5 1.01
1 1.006
1.003
1.001
I .ooo
77.380
-
1.378
1.250
1.183
1.142
1.115
1.095
1.080
1.068
1.059
1.04
6 1.037
1.030
1.025
1.02
1
1.016
1.01 1
1.007
1.003
1.001
1 .ooo
82.292
-
1.3%
1.260
1.190
1.147
1.1
18
1.098
1.082
1.07
0 1.061
1 .a7
1.038
.03 1
.026
.022
.016
.01 I
.007
.003
1.001
1 .OO
o
85.749
1.587
1.329
I .227
1.171
1.135
1.110
I .092
1.078
1.067
I .058
1.04
6 1.037
1.030
1.025
1.022
1.016
1.011
1.00
7 1.003
1.001
1 .Ooo
74.3970
1.643
1.350
1.239
1.179
1.141
1.1 14
1.095
1.08 1
1.070
1.061
1.047
1.038
1.03 1
1.026
1.022
1.017
1.012
1.007
1.003
1.001
1 .O
oo
79.08 19
I .697
1.370
1.25 1
1.186
1.146
1.119
1.099
1.084
1.072
1.063
1.049
1.039
1.032
1.027
1.023
1.017
1.012
1.007
1.004
1.001
1 .Ooo
83.2977
1.768
1.3%
1.265
1.1%
1.153
1.124
1.103
1.087
1.075
1.065
1.05 1
1.041
1.033
1.028
1.024
1.018
1.012
1.00
8 1.
004
I .001
I .ooo
88.3794
1.821
1.415
1.275
1.203
1.158
1.128
1.106
1.089
1.077
1.067
I .052
1.042
1.034
1.029
1.024
1.018
1.013
1.008
1.004
1.001
I .Ooo
9 1.95 1
7
r=1
3
r=1
4
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.0
25
0.01
0 0.
005
1-
2
1.34
5 3
1.24
1 4
1.18
2 5
1.14
5
6 1.
118
7 1.
099
8 1.
084
9 1.
073
10
1.06
4
12
1.05
0 14
1.
040
16
1.03
3 I8
1.
028
20
1.02
4
24
1.01
8 30
1.
013
40
1.00
8 60
1.00
4 12
0 1.
001
m
1.OOo
-
1.36
7 1.
253
1.19
1 1.
151
1.12
3 1.
103
1.08
8 1.
076
1.06
6
1.05
2 1.
042
1.03
5 1.
029
1.02
5
1.01
9 1.
013
I .00
8 1.
004
1.00
1
1 .Ooo
-
I .387
1.2
65
1.19
9 1.
157
1.12
8 1.
107
1.09
1 1.
078
1.06
8
I .05
4 I .0
43
1.03
6 1.
030
1.02
6
1.01
9 1.
013
1.00
8 1.
004
I .oOl
1 .Ooo
-
1.41
4 1.
280
1.208
1.
164
1.13
3 1.
111
1.09
4 1.
08 I
1.07
I
I .05
5 1.
045
1.03
7 I .0
3 1
I .026
1.02
0 1.
014
1.00
9 1.
004
1.00
1
1 .Ooo
-
1.43
3 1.
290
1.21
5 1.
169
1.13
7 1.1
14
I .097
1.
083
I .073
1.05
7 1.
046
1.03
8 1.
032
I .027
1.02
0 1.
014
1.00
9 1.
004
1.00
1
1 .OOo
xf;,
79.9
73
84.8
21
89.1
77
94,4
22
98.1
05
1.626
1.
361
1.25
4 1.
194
1.155
1.127
1.
107
1.09
1 1.
079
1.06
9
1.05
5 1.
044
1.037
I .0
3 1
I .026
1.02
0 1.0
14
1.00
9 1.
004
1.00
1
I .Ooo
1.68
3 1.3
83
1.26
7 1.
203
1.16
1
1.13
2 1.1
11
1.095
1.
082
1.07
2
1.05
7 1.
046
1.03
8 1.
032
1.027
1.02
1 1.
014
I .009
1.
004
1.00
1
1 .Ooo
1.740
I .404
1.27
9 1.
21 1
1.
167
1.13
7 1.
1 I5
I .098
I .0
85
1.07
4
1.05
8 1.
047
I .039
1.
033
I .028
1.02
1 1.0
15
1.00
9 1.0
05
1.00
1
1 .Ooo
1.81
3 1.
431
1.29
4 1.
22 1
1.
174
1.14
3 1.
1 I9
1.
102
1.08
8 1.
077
1.06
0 1.0
49
1.04
0 1.
034
1.02
9
1.02
2 1.
015
1.01
0 1.
005
1.00
1
1 .Ooo
1.86
7 I .4
5 1
1.30
5 1.
228
1.18
0
1.14
7 1.
123
1.10
4 I .o
m
1.07
9
1.06
2 1.
050
1.04
1 1.
035
1.03
0
1.02
2 1.
016
1.010
1.
005
1.00
1
1 .OOo
85.5
270
90.5
312
95.0
232
100.
4252
10
4.21
49
m
N
0
Tab
le9
(Cm
rin
ucd
)
m =
5 -
r=1
5
r=1
6
a
0.10
0 0.050
0.02
5 0.
010
0.00
5 0.
0100
0.050
0.01
0 0.
005
1-
2
1.37
7 3
1.26
7 4
1.20
5 5
1.16
4
6 1.
136
7 1.
115
8 1.
098
9 1.
085
10
1.07
5
12
1.05
9 14
1.
048
16
1.04
0 18
1.
034
20
1.02
9
24
1.02
2 30
1.
015
40
1.01
0 60
1.
005
120
1.00
1
00
1.oo
o
x:,
91.0
61
-
1.39
9 1.
281
1.21
4 1.
171
1.14
1 1.
119
1.10
2 1.
088
1.07
8
1.06
1 1.
050
1.04
1 1.
035
1 .OM
1.02
3 1.
016
1.01
0 I .005
1.00
1
1 .O
oo
96.2
17
-
1.42
1 1.
293
1.22
3 1.
177
1.14
6 1.
123
1.10
5 1.
091
1.08
0
1.06
3 1.
051
I .04
3 1.
036
1.03
1
1.02
3 1.
016
1.01
0 1.
005
I .ool
1 .OOo
100.
839
-
1.44
9 1.
309
1.23
3 1.
185
1.15
2 1.
127
1.10
9 1 .o
w
1.08
3
1.06
5 1.
053
1.04
4 I .0
37
1.03
2
1.02
4 1.
017
1.01
1 I .
005
1.00
1
1 .m
106.
393
-
1.46
9 1.
320
1.24
0 1.
190
1.15
6 1.
131
1.1 1
2 1.
097
I .08
5
1.06
7 1 .OM
1.04
5 1.
038
1.03
3
1.02
5 1.
017
1.01
1
1.00
5 I .
002
1 .Ooo
1 10.
286
1.66
3 1.
392
1.28
0 1.
216
1.17
4
1.14
4 1.
122
1.10
5 1.
091
1 .os 1
1.06
4 1.
052
1.04
3 1.
037
1.03
2
I .024
1.
017
1.01
1 1.
005
1.00
2
1 .Ooo
1.72
2 1.
415
1.29
4 I .
226
1.18
1
1.15
0 .1
27
.I09
.0
95
.083
.066
.ow
.045
1.
038
1.03
3
1.025
1.
018
1.01
1 1.
005
1 .m
2
1 .Ooo
1.78
0 1.
437
1.30
7 1.
234
1.18
8
1.15
5 1.
131
1.1
12
1.09
8 1.
086
1.06
8 1.
055
1.04
6 1.
039
1.03
3
I .02
5 1.
01s
1.01
1 1.
006
1.00
2
1 .Ooo
1.85
5 1.
465
1.32
3 1.
245
1.19
5
1.16
1 .1
36
.116
.I01
.089
.070
.0
57
.048
I .040
1.03
5
I .02
6 1.
01 9
1.01
2 ! .006
1.00
2
1 .OOo
1.91
1 1.
486
1.33
4 I .
253
1.20
1
1.16
5 1.
139
1.11
9 1.
104
1.09
1
1.07
2 1.
059
I .04
9 1.
041
1.03
5
1.02
7 1.
019
1.01
2 1.
006
1.00
2
1 .ooo
96.5
782
101.
8795
106
.628
6 11
2.32
88 1
16.3
21 1
r=1
7
r=1
8
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
-
I 2 1.
407
3 1.
293
4 1.
227
5 1.
184
6 1.
153
7 1.
130
8 1.
112
9 1.
098
10
1.08
6
12
14
16
18
20
24
.069
,0
56
,047
.0
40
.034
,026
30
1.
019
40
1.01
2 GO
1.
006
120
1.00
2
00
1.OOo
-
1.43
1 I .3
07
1.23
7 1.
191
1.15
9 1.
134
1.1 1
6 1.
101
I .089
.07 1
,0
58
.048
.04 I
.0
35
.027
1.
019
1.01
2 1.
006
1.00
2
1 .ooo
-
1.45
3 1.
320
1.24
6 1.
198
1.16
4 1.
139
1.11
9 1.
104
I .092
,073
.060
,050
.0
42
.036
.028
I .0
20
1.01
2 1.
006
1.00
2
1 .OOo
- 1.48
2 1.
336
1.25
7 I .2
06
1.17
0 1.
144
1.12
4 1.
I08
1.09
5
1.07
5 1.
062
1.05
1 1.
044
1.03
7
1.02
9 1.
020
1.01
3 1.
006
I .002
I .Ooo
-
1 SO
3 1.
348
1.26
5 1.
212
1.17
5 1.
147
1.12
7 1.
110
1.09
7
.077
.0
63
,052
,044
.038
.029
1.
02 1
1.01
3 1.
006
1.00
2
1 .Ooo
7
XL
10
2.07
9 10
7.52
2 11
2.39
3 11
8.23
6 12
2.32
5
1.69
8 1.
421
1.30
5 1.
238
1.19
3
1.16
1 1.
137
1.1 1
9 1.
104
1.09
2
I .073
I .060
1.05
0 I .0
43
1.03
7
1.02
8 1.
020
1.01
3 1.
006
1.00
2
1 .Ooo
1.75
8 1.
445
1.32
0 1.
248
1.20
1
1.16
7 1.
142
1.12
3 I.
I07
1.09
5
I .076
1.
062
1.05
2 1.
044
1.03
8
I .029
I .0
2 I
1.01
3 1.
007
I .002
I .Ooo
1.81
8 I .468
1.33
3 1.
257
I .20
8
1.17
3 1.
147
I. I2
6 1.
110
1.09
8
1.07
8 1.
064
I .053
1.
045
1.03
9
1.03
0 1.
02 1
1.01
3 1.
007
1.00
2
I .Ooo
1.89
5 1.
498
1.35
0 1.
268
1.21
6
1.17
9 1.
152
1.13
1 1.
114
1.10
1
1.08
0 I .0
66
1.05
5 1.
047
1.04
0
1.03
1 1.
022
1.01
4 I .0
07
1.00
2
1 .Ooo
1.95
3 1.
519
1.36
2 1.
277
I .22
2
1.18
4 1.
156
1.13
4 1.
117
1.10
3
1.08
2 I .
067
1.05
6 1.
048
1.04
1
1.03
1 I .0
22
1.01
4 I .
007
1.00
2
I .OOo
10
7.56
50 1
13.1
453
118.
1359
124
.116
3 12
8.29
89
m
N
N
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
120 m
X5,
~~
~~~
~~
~~~
m=
5 -
r=1
9
1 =
20
a
0.10
0 0.050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.005
1.79
3 1.
853
-
1.43
6 1.
318
1.24
9 1.
203
1.17
0 1.
145
1.12
6 1.
110
1.09
7
1.07
8 1.
064
1.05
4 1.
046
1.04
0
1.03
1
1.02
2 1.
014
1.00
7 I .0
02
I .Ooo
11
3.03
8
-
1.46
0 1.
332
1.25
9 1.
210
1.17
6 1.
150
1.13
0 1.
114
1.10
1
1.08
1 1.
066
1.05
6 1.
047
1.04
1
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .ax,
1 18.
752
-
1.48
3 1
.34
1.
268
1.21
7
1.18
1 1.
154
1.13
4 1.
117
1.10
3
1.08
3 1.
068
1.05
7 1.
049
1.04
2
1.03
2 1.
023
1.01
5 1.
007
1.00
2
1 .Ooo
12
3.85
8
-
1.51
3 1.
363
1.28
0 1.
226
1.18
8 1.
160
1.13
8 1.
121
1.10
7
1.08
5 1.
070
1.04
9 1.
050
1.04
3
1.03
3 1.
024
1.01
5 1.
008
1 .#2
1 .Ooo
129.
973
-
1.53
5 1.
375
1.28
8 1.
232
1.19
3 1.
164
1.14
1 1.
124
1.10
9
1.08
7 1.
072
1.06
0 1.
05 1
1.04
4
1.03
4 1.
024
1.01
5 I .
008
1.00
2
1 .Ooo
134.
247
i.73
i 1.
449
1.33
0 1.
259
1.21
2
1.17
8 1.
152
1.13
2 1.
116
1.10
3
1.08
3 1.
069
1.05
8 1.
049
1.04
3
1.03
3 1.
024
1.01
5 1.
008
1.00
2
1 .Ooo
1.47
4 1.
345
1.27
0 1.
220
1.18
4 1.
157
1.13
7 1.
120
1.10
6
1.08
6 1.
07 1
1.05
9 1.
05 1
1.04
4
1.03
4 1.
024
1.01
5 1.
008
1.00
2
1 .Ooo
1.49
8 1.
358
1.27
9 1.
227
1.19
0 1.
162
1.14
1 1.
123
1.10
9
1.08
8 1.
072
1.06
1 1.
052
1.04
5
1.03
5 1.
025
1.01
6 1.
00s
1.00
2
1 .OOo
1.93
3 1.
528
1.37
6 1.
291
1.23
6
1.19
7 1.
168
1.14
5 1.
127
1.1
13
1.09
1 1.
075
1.06
3 1.
053
1.04
6
1.03
6 1.
025
1.01
6 1.
008
1.00
2
1 .Ooo
~
1.99
2 1.
55 1
1.
388
1.30
0 1.
242
1.20
2 1.
172
1.14
9 1.
130
1.1
15
1.09
2 1.
076
1.06
4 1.
054
1.04
7
1.03
6 1.
026
1.01
6 1.
008
1.00
2
1 .Ooo
118.
4980
124
.342
1 12
9.56
12
135.
8067
140
.169
5
m =
6 -
r =
6 r=
7
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
120
a,
1.47
1 1 .
237
1.15
3 1.
109
1.08
3
1.06
6 1.
053
1.04
4 1.
037
1.03
2
1.02
4 1.
019
1.01
5 1.
013
1.01
I
I .008
I .0
05
1.00
3 I .0
02
1 .Ooo
I .Ooo
1.52
0 1.
255
1.16
3 1.
116
1.08
8
1.06
9 1.
056
1.04
6 1.
039
1.03
4
1.02
5 I .0
20
1.01
6 1.
013
1.01
1
I .00
8 I .006
1.00
3 1.
002
1 .Ooo
I .Ooo
1.56
8 1.
272
1.17
2 1.
122
1.09
2
1.07
2 1.
058
I .048
1.
041
I .035
1.02
6 1.
02 1
1.01
7 1.
014
1.01
2
1.00
9 I .006
1.00
3 I .0
02
1 .Ooo
1 .OOo
1.63
1 1.
294
1.18
3 1.
129
1.09
7
1.07
6 1.
061
1.05
1 1.
043
1.03
7
1.02
8 1.
022
1.01
8 1.
014
1.01
2
1.00
9 I .0
06
1.00
4 1.
002
1 .ooo
1 .Ooo
1.67
7 1.
310
1.19
2 1.
134
1.10
1
1.07
9 1.
064
I .05
3 I .
044
I .038
1.02
9 1.
022
1.01
8 1.
015
1.01
3
1.00
9 I .006
I .00
4 1.
002
1 .Ooo
1 .Ooo
47
.212
2 50
.998
5 54
.437
3 58
.619
2 61
.581
2
1.48
1 1.
249
1.16
3 1.
118
1.09
0
1.07
2 1.
059
1.04
9 1.
042
1.03
6
1.02
7 1.
022
1.10
8 1.
014
1.01
2
1.00
9 I .0
06
1.00
4 I .0
02
I .OOo
1 .O
oo
I .53
0 1.
266
1.17
3 1.
124
I .09
5
1.07
5 1.
062
1.05
1 1.
043
1.03
7
1.02
9 1.
023
1.01
5 1.
013
1.00
9 I .006
1.00
4 I .0
02
1 .Ooo
I .O
Oo
1 .O
W
1.57
9 1.
284
1.18
2 1.
131
1.09
9
1.07
9 1.
064
1.05
3 1.
045
1.03
9
1.03
0 1.
023
1.01
9 1.
016
1.01
3
1.01
0 1.
007
I .004
1.
002
1.00
1
1 .Ooo
1.64
2 1.
306
1.19
4 1.
138
1.10
5
1.08
3 I .
067
1.05
6 I .
047
1.04
1
1.03
1 1.
024
1.02
0 1.
016
1.01
4
1.01
0 1.
007
1.00
4 1.
002
1.00
1
1 .Ooo
1.68
8 1.
322
1.20
3 1.
144
1.10
9
1.08
6 1.
070
1.05
8 1.
049
1.04
2
1.03
2 I .
025
1.02
0 1.
017
1.01
4
1.01
0 1.
007
1.00
4 1.
002
1.00
1
1 .Ooo
54
.090
2 58
.124
0 61
.776
8 66
.206
2 69
.336
0
Tab
le 9
(C
mth
ue
d)
~ ~~
~~~~
m=
6 -
r =8
r
=9
u
0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
I 1.
494
1.54
3 1.
592
1.65
6 1.
703
1.67
1 1.
719
2 1.
261
3 1.
174
4 1.
127
5 1.
098
6 1.
079
7 1.
065
8 1.
054
9 1.
046
10
1.04
0
12
1.03
1 14
1.
024
16
1.02
0 18
1.
016
20
1.01
4
24
1.01
0 30
1.
007
40
1.00
4 60
1.
002
I20
1.00
1
00
1.OOo
1.27
9 1.
297
1.18
4 1.
194
1.13
4 1.
140
1.10
3 1.
108
1.08
2 1.
086
1.06
8 1.
070
1.05
7 1.
059
1.04
8 1.
050
1.04
2 1.
043
1.03
2 1.
033
1.02
5 1.
026
1.02
1 1.
021
1.01
7 1.
018
1.01
4 1.
015
1.01
1 1.
011
1.00
7 1.
008
1.00
4 1.
005
1.00
2 1.
002
1.00
1 1.
001
1.OOo
1.O
Oo
1.31
9 I .
205
1.14
8 1.
113
1 .ow
1.
074
I .06
2 1.
052
1.04
5
1.03
5 1.
027
1.02
2 1.
108
1.01
5
1.01
1 1.
008
1.00
5 1.
002
1.00
1
1 .OOo
2
X,,,
60.9
066
65.1
708
69.0
226
73.6
826
76.%
88
1.33
6 1.
214
1.15
3 1.
1 17
1.09
3 1.
076
1.06
3 1 .OM
I .046
1.03
6 1.
028
1.02
2 1.
019
1.01
6
1.01
2 1.
008
I .00
5 1.
002
1.00
1
1 .Ooo
1.50
8 1.
275
1.18
5 1.
137
1.10
7
1.08
6 1.
07 1
1.06
0 1.
051
1.04
4
I .03
4 1.
027
1.02
2 1.
019
1.01
6
1.01
2 1.
008
1.00
5 1.
002
1.00
1
1 .Ooo
~
1.55
8 1.
607
1.29
3 1.
196
1.14
4 1.
1 12
1 .ow
1.
074
I .06
2 1.
053
1.04
6
1.03
5 1.
028
1.02
3 1.
019
1.01
6
1.01
2 1.
008
1.00
5 1.
002
1.00
1
1 .Ooo
1.31
I 1.
205
1.15
0 1.
116
1.09
3 1.
077
I .06
5 1.
055
1.04
8
1.03
6 1.
209
1.02
4 I .
020
1.01
7
1.01
3 1.
009
1.00
5 I .0
03
1.00
1
1 .OOo
1.33
3 1.
350
1.21
8 1.
227
1.15
8 1.
164
1.12
2 1.
126
1.09
8 1.
101
1.08
0 1.
083
1.06
7 1.
069
1.05
8 1.
059
1.05
0 1.
051
1.03
7 1.
037
1.03
1 1.
031
1.02
4 1.0
25
1.02
0 1.
021
1.01
7 1.
018
1.01
3 1.
013
1.00
9 1.
009
1.00
5 1.
006
1.00
3 1.
003
1.00
1 1.
002
1.ooo
1.O
Oo
67.6
728
72.1
532
76.1
921
81.0
688
84.5
016
r=lO
r=
12
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 1.
523
2 1.
288
3 1.
197
4 1.
147
5 1.
115
6 1.
093
7 1.
078
8 1.
066
9 1.
056
10
1.04
9
12
1.03
8 14
1.
031
16
1.02
5 18
1.
021
20
1.01
8
24
1.01
3 30
1.
009
40
1.00
6 60
1.
003
120
1.00
1
00
1.OOo
1.57
3 1.
307
1.20
8 1.
154
1.12
0
1.09
7 1.
08 I
1.06
8 I .
059
1.05
1
1.04
0 1.
032
1.02
6 1.
021
1.01
8
1.01
4 1.
010
1.00
6 1.
003
1.00
1
1 .Ooo
~
1.62
3 1.
325
1.21
8 1.
161
1.12
5
1.10
1 1 .o
w 1.
07 1
1.06
1 1.
053
1.04
1 1.
033
1.02
6 1.
022
1.01
9
1.01
4 1.
010
1.00
6 1.
003
1.00
1
1 .Ooo
1.68
7 1.
348
1.23
0 1.
169
1.13
1
1.10
6 1.
087
1.07
4 1.
063
1.05
5
1.04
2 I .0
34
1.02
7 1.
023
1.01
9
1.01
5 1.
010
1.00
6 1.
003
1.00
1
1 .Ooo
1.73
6 1.
365
I .239
1.
175
1.13
5
1.10
9 I .090
1.07
6 1.
065
I .056
1.04
3 1.
034
1.02
8 1.
023
I .020
1.01
5 1.
010
1.00
6 1.
003
1.00
1
1 .OOo
xf
, 74
.397
0 79
.081
9 83
.297
6 88
.379
4 91
.951
7
1.55
4 1.
316
1.22
1 1.
167
1.13
3
1.10
9 1.
091
1.07
8 1.
067
1.05
9
1.04
6 1.
037
1.03
1 1.
026
1.02
2
1.01
7 1.
012
1.00
7 I .0
04
1.00
1
1 .Ooo
1.60
5 1.
335
1.23
2 1.
175
1.13
8
1.11
3 1.
095
1.08
1 1.
070
1.06
1
I .048
1.03
9 1.
032
1.02
7 1.
023
1.01
7 1.
012
1.00
7 1.
004
1.00
1
1 .OOo
1.65
5 1.
354
I .242
1.
182
1.14
4
1.11
7 1.
098
1.08
3 1.
072
1.06
3
1.04
9 1.
040
1.03
3 1.
027
1.02
3
1.01
8 1.
012
1.00
8 1.
004
1.00
1
1 .Ooo
I .722
1.
378
I .255
1.
190
1.15
0
1.12
2 1.
102
1.08
6 I .0
74
1.06
5
1.05
1 1.
041
1.03
4 1.
028
1.02
4
1.01
8 1.
013
1.00
8 1.
004
1.00
1
1 .Ooo
1.77
1 1.
395
1.26
5 I.
197
1.15
4
1.12
5 1.
104
1.08
9 1.
076
1.06
7
1.05
2 I .
042
1.03
5 1.
029
1.02
5
1.01
9 1.
013
1.00
8 1.
004
1.00
1
1 .Ooo
87.7
430
92.8
083
97.3
53 I
102.
8163
106
.647
6
Tab
le9
(Con
tin
urd
) ~
~ ~~
~
m =6 -
r=14
r=15
0.050
0.025
0.010
0.005
0.100
0.050
0.025
0.010
0.005
1 1.585
2 1.343
3 1.
244
4
1.188
5 1.151
6 1.125
7 1.105
8 1.
090
9 1.078
10
1.069
12
1.055
14
1.044
16
1.037
18
1.031
20
1.027
24
1.020
30
1.014
40
1.009
60
1.00
4 120
1.001
Q:
1.OOo
1.637
1.363
1.25
6 1.1
% 1.157
1.129
1.109
1.093
1.081
1.07 1
1.056
1.04
6 1.038
1.032
I .028
1.02 1
1.015
1.00
9 1.005
I .001
1 .ooo
1.688
1.756
1.383
1.407
1.267
1.281
1.203
1.212
1.162
1.169
1.133
1.139
1.112
1.117
1.096
1.100
1.083
1.08
6 1.073
1.076
1.058
1.060
1.047
1.049
1.039
1.040
1.033
1.034
1.028
1.029
1.021
1.022
1.01
5 1.016
1.00
9 1.
010
1.005
1.005
1.001
1.001
1.OOo
1.Ooo
I .806
I .425
1.291
1.219
1.174
1.142
1.119
1.102
I .088
1.078
1.06
1 1.050
1.041
1.035
1.030
1.023
1.016
1.010
1.005
1.001
I .ooo
x;,
100.9800 1M.3948 1
1 1.2423 117.0565 121.1263
-
1.357
1.256
1.198
1.160
1.132
1.1 12
1.097
I -084
1.074
1.059
1.048
1.040
1.034
1.029
1.022
1.016
1.010
1 .005
1.001
1 .ooo
107.565
-
1.377
1.268
1.206
1.16
6
1.137
1.1 16
1.100
1.087
1.076
1.061
1.050
1.041
1.035
1.030
1.023
1.016
1.010
1.005
1.001
1 .ooo
113.145
-
1.397
1.279
1.214
1.171
1.142
1.120
1.103
1 .089
1.079
1.062
1.051
1.042
1.036
1.03 1
1 .ow
1.017
1.010
1.005
1.001
1 .OOo
118.136
-
1.422
1.293
1.223
1.178
1.147
1.124
1.106
1 -09
2 1.08 1
1.06
4 1.053
1.044
1.037
1.032
1.024
1.017
1.01 1
1.005
1.002
1 .Ooo
124.1 16
-
1.44
0 1.303
1.230
1.183
1.151
1.127
1.109
1.095
1.08
3
1.06
6 1.054
1.045
1.038
1.032
1.025
1.017
1.01
1 1.005
1 .OGt
1 .o
oo 128.299
r=1
6
r= 1
7 a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1 1.
615
2 1.
370
3 1.
267
4 1.
208
5 1.
168
6 1.
140
7 1.
119
8 1.
103
9 1.
090
10
1.07
9
12
1.06
3 14
1.
052
16
1.04
3 18
1.
037
20
1.03
2
24
1.02
4 30
1.
017
40
1.01
1 60
1.
005
120
1.00
2
00
1.OOo
I .668
1.
391
1.28
0 1.
216
1.17
5
I. I4
5 1.
123
1.10
6 I .
093
1.08
2
1.06
5 1.
053
1.04
5 1.
038
1.03
3
1.02
5 1.
018
1.01
1 1.
006
1.00
2
1 .OOo
1.72
1 1.4
1 I
1.29
1 1.
224
1.18
1
1.15
0 1.
127
1.10
9 I .0
95
I .OM
I .06
7 1.
055
1.04
6 1.
039
I .033
I .025
1.
018
1.01
1 1.
006
I .002
1 .Ooo
1.78
9 1.
436
1.30
5 1.
234
1.18
8
1.15
5 1.
131
1.1
13
I .099
1.08
7
I .06
9 1.
056
1.04
7 1.
040
I .034
I .026
1.
019
1.01
2 1.
006
1.00
2
1 .Ooo
1.84
1 I .544
1.31
6 1.
241
1.19
3
1.15
9 1.
135
1.11
6 1.
101
1 .08
9
1.07
1 I .0
58
1.04
8 1.
041
1.03
5
1.02
7 1.
019
1.01
2 1.
006
1.00
2
1 .Ooo
x:,,~
114
.130
7 11
9.87
09 1
25.0
001
131.
1412
135
.433
0
-
1.38
3 1.
279
1.21
8 1.
177
1.14
8 1.
126
1.10
9 I .O
% 1.
085
1.06
8 1.
056
1.04
7 1.
040
1.03
4
1.02
6 1.
019
1.01
2 I .0
06
1.00
2
1 .Ooo
120.
679
-
I .404
1.29
1 1.
226
1.18
4
1.15
3 1.
130
1.11
3 1.
099
1.08
7
1.07
0 I .0
57
I .04
8 1.
041
1.03
5
1.02
7 1.
019
1.01
2 1.
006
I ,00
2
1 .Ooo
126.
574
-
I .424
1.
303
1.23
4 1.
190
1.15
8 1.
134
1.11
6 1.
101
1 .ow
I .072
1.
059
1.04
9 1.
042
1.03
6
1.02
8 1.
020
1.01
2 I .cQ6
1.00
2
1 .Ooo
-
1.45
0 1.
317
1.24
4 1.
197
1.16
4 1.
139
1.12
0 1.
105
1.09
2
1.07
4 1.
061
1.05
1 I .0
43
1.03
7
1.02
8 1.
020
1.01
3 I .0
06
I .002
1 .00
0
-
I .469
1.
328
1.25
1 1.
202
1.16
8 1.
142
1.12
3 1.
107
1.09
4
1.07
5 1.
062
1.05
2 1.
044
1.03
8
1.02
9 1.
02 1
1.01
3 1.
007
I .00
2
1 .Ooo
131.
838
138.
134
142.
532
Tab
le9
(Co
nti
mu
d)
rn =
€ -
r=1
8
r=1
9
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1.69
8 1.
752
1.82
2 1.
874 -
1.41
7 1.
303
1.23
7 1.
193
1.16
1 1.
138
1.1
19
1.10
5 1.
093
1.07
4 1.
061
1.05
1 1.
044
1.03
8
1.02
9 1.
02 1
1.01
3 1.
007
1.00
2
I .Ooo
1.43
8 1.
464
1.31
5 1.
329
1.24
5 1.
255
1.19
9 1.
206
1.16
6 1.
172
1.14
2 1.
146
1.12
3 1.
127
1.10
7 1.
111
1.09
5 1.
098
1.07
6 1.
079
1.06
3 1.
065
1.05
3 1.
0%
1.04
5 1-
046
1.03
9 1.
040
1.03
0 1.
031
1.02
1 1.
022
1.01
3 1.
014
1.00
7 1.
007
1.00
2 1.
002
1.OOo
1.
Ooo
1.48
3 1.
340
1.26
2 1.
212
1.17
6 1.
150
1.12
9 1.
113
1.10
0
I .08
0 1.
066
1.05
5 1.
047
1.04
1
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .ooo
1 1.
644
2 1.
3%
3 1.
290
4 1.
228
5 1.
186
6 1.
156
7 1.
133
8 1.
116
9 1.
101
10
1.09
0
12
1.07
2 14
1.
060
16
1.05
0 18
1.
043
20
1.03
7
24
1.02
8 30
1.
020
40
1.01
3 60
1.
006
120
1.00
2
a2
1.OO
o
-
1.40
8 1.
301
1.23
7 1.
195
1.16
4 1.
140
1.12
2 1.
107
1.09
5
1.07
7 1.
063
1.05
3 1.
046
1.03
9
1.03
0 1.
022
1.01
4 1.
007
1 .oc
2
1 .Ooo
-
1.43
0 1.
314
I .246
1.
201
1.16
9 1.
145
1.12
6 1.
1 10
1.
098
1.07
9 I .
065
1 .05
5 1.
047
1.04
1
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .ooo
-
1.45
1 1.
326
1.25
5 1.
208
1.17
4 1.
149
1.12
9 1.
113
1.10
1
1.08
1 I .
067
1.05
6 1.
048
1.04
1
1.03
2 1.
023
1.01
5 1.
007
1.00
2
1 .O
oo
-
1.47
7 1.
497
1.34
1 1.
352
1.26
5 1.
273
1.21
5 1.
221
1.18
0 1.
184
1.15
4 1.
157
1.13
3 1.
136
1.11
7 1.
119
1.10
4 1.
106
1.08
3 1.
085
1.06
9 1.
070
1.05
8 1.
059
1.04
9 1.
050
1.04
3 1.
043
1.03
3 1.
033
1.02
3 1.
024
1.01
5 1.
015
1.00
8 1.
008
1.00
2 1.
002
1.OOo
1.
OO
o
x:m
127
.21 1
1 13
3.25
69 1
38.6
506
145.
0988
149
.599
4 13
3.72
9 13
9.92
1 14
5.44
1 15
2.03
7 15
6.63
7
r=2
0
u
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
1 20
00
~
1.67
2 1.
727
1.78
1 1.
420
1.44
3 1.
464
1.31
2 1.
325
1.33
7 1.
247
1.25
6 1.
265
1.20
3 1.
210
1.21
7
1.17
1 1.
177
1.18
2 1.
147
1.15
2 1.
156
1.12
8 1.
132
1.13
6 1.
113
1.11
6 1.
119
1.10
1 1.
103
1.10
6
1.08
1 1.
084
1.08
6 1.
067
1.06
9 1.
071
1.05
7 1.
058
1.06
0 1.
049
1.05
0 1.
051
1.04
2 1.
043
1.04
4
1.03
3 1.
033
1.03
4 1.
023
1.02
4 1.
025
1.01
5 1.
015
1.01
6 1.
008
1.00
8 1.
008
1.00
2 1.
002
1.00
2
1.Ooo
1.O
oo
1.ooo
1.85
3 1.
490
1.35
3 1.
275
1.22
4
1.18
8 1.
161
1.14
0 1.
123
1.10
9
I .08
8 I .0
73
1.06
1 1.
052
1 .04
5
1.03
5 1.
025
1.01
6 1.
008
1.00
2
1 .Ooo
I .906
1.
510
1.36
4 I .
283
1.23
0
1.19
3 1.
165
I. 14
3 1.
126
1.1
I1
1 .ow
1.07
4 1.
062
1.05
3 1.
046
I .03
6 1.
026
1.01
6 1.
008
1.00
2
1 .Ooo
x:,,
140.
2326
14
6.56
74
152.
21 14
15
8.95
02
163.
6482
Tab
le9
(Co
nth
ud
)
I 1.
484
1.53
2 2
1.25
6 1.
273
3 1.
170
1.18
0 4
1.12
5 1.
131
5 1.
096
1.10
1
6 1.
077
1.08
1 7
1.06
3 1.
066
8 1.
053
1.05
5 9
1.04
5 1.
047
10
1.03
9 1.
041
12
1.03
0 1.
031
14
1.02
4 1.
025
16
1.01
9 1.
020
18
1.01
6 1.
017
20
1.01
4 1.
014
24
1.01
0 1.
010
30
1.00
7 1.
007
40
1.00
4 1.
004
60
1.00
2 1.
002
120
1.00
1 1.
001
00
1.Ooo
1.O
Oo
1.38
0 1.
643
1.29
0 1.
312
1.18
9 1.
201
1.13
7 1.
145
1.10
5 1.
111
1.08
4 1.
088
1.06
9 1.
072
1.05
8 1.
060
1.04
9 1.
051
1.04
2 1.
044
1.03
2 1.
034
1.02
6 1.
027
1.02
1 1.
022
1.01
7 1.
018
1.01
5 1.
015
1.01
1 1.
011
1.00
7 1.
008
1.00
4 1.
005
1.00
2 1.
002
1.00
1 1.
001
1.Ooo
1.O
oo
1.68
9 1.
329
1.21
0 1.
150
1.11
4
1.09
1 I .
074
1 .M
2 1.
053
1.04
5
1.03
5 1.
028
1.02
2 1.
019
1.01
6
1.01
2 I .
008
1.00
5 1.
002
1.00
1
1 .Ooo
m =7 -
r=7
r=
8 a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1.49
0 1.
538
1.586
1.
648
1.69
4 1.
265
1.28
2 1.
179
1.18
9 1.
132
1.13
9 1.
103
1.10
8
1.08
3 1.
086
1.06
8 1.
071
1.05
8 1.
060
1.04
9 1.
051
1.04
3 1.
044
1.03
3 1.
034
1.02
6 1.
027
1.02
1 1.
022
1.01
8 1.
018
1.01
5 1.
016
1.01
1 1.
012
1.00
8 1.
008
1.00
5 1.
005
1.00
2 1.
002
1.00
1 1.
001
1.O
oo
1.OOo
1.29
9 1.
321
1.19
8 1.
210
1.14
5 1.
152
1.11
2 1.
117
1.09
0 1.0
94
1.07
4 1.
077
1.06
2 1.
065
1.05
3 1.
055
1.04
6 1.
048
1.03
5 1.
037
1.02
8 1.
028
1.02
3 1.
023
1.01
9 1.
020
1.01
6 1.
017
1.01
2 1.
012
1.00
8 1.
009
1.005
1.
005
1.00
2 1.
003
1.00
1 1.
001
1.Ooo
1.O
Oo
1.33
7 1.
218
1.15
8 1.
121
I .09
7 1.
080
I .06
7 I .0
57
1.04
9
1.03
8 1.
029
1.02
4 1.
020
1.01
7
1.01
3 1.
009
1.00
5 1.
003
1.00
1
1 .Ooo
xt,,,
62.0
375
66.3
386
70.2
224
74.9
195
78.2
307
69.9
185
74.4
683
78.5
672
83.5
134
86.9
938
r =
9
r=lO
01
0.100
0.050
0.025
0.01
0 0.005
0.100
0.050
0.025
0.010
0.005
I 1.499
I .547
1.594
1.656
1.703
1 so9
1.557
2 1.275
3 1.188
4 1.140
5 1.110
6 1.089
7 1.074
8 1.063
9 1.
054
10
1.047
12
1.036
14
1.029
16
1.024
18
1.020
20
1.017
24
1.013
30
1.009
40
1.005
60
1.003
120
1.001
oc
1.Ooo
1.292
1.198
1.147
1.1 15
I .093
1.077
I .065
1.056
1.048
1.038
1.030
1.025
1.02 1
1.018
1.013
1.009
1.006
1.003
1 .001
I .Ooo
1.309
1.207
1.153
1.1 19
I .096
1 .080
1.067
1.058
1.05
0
1.039
1.03 1
I .025
1.021
1.018
1.013
1.00
9 1.006
1.003
1.001
1 .Oo
o
1.33 1
1.219
1.161
1.125
1.100
I .083
1.070
1.06
0 1.052
1 .w
1.032
1.026
1.022
1.019
1.014
1.010
1.00
6 1.003
1.001
1 .Ooo
1.348
1.228
1.16
6 1.129
1.103
1.085
I .072
1.062
I .053
.04 1
,033
.027
.023
.019
.014
1.010
I .006
1.003
1 .001
1 .Ooo
1.285
1.197
1.148
1.117
I .o%
1.080
1.068
1.058
1.05 1
1.
040
1.032
1.026
1.022
1.019
1.014
1.010
1.00
6 1.003
1.001
I .O
oo
1.303
1.208
1.155
1.122
1.099
1.083
1.070
I .060
1.053
I .042
1.034
1.028
1.023
1.019
1.014
1.010
1.006
1.003
1.001
I .O
oo
1.60
4 1.320
1.217
1.162
1.127
1. I03
I .086
1.073
1.062
1.054
1.042
1.034
1.028
1.023
1.020
1.01
5 1.010
1.006
1.003
1 .ool
1 .Ooo
x:,,,
77.7454
82.5287
86.82%
92.0100
95.6493
1.666
1.342
I .229
1.169
1.132
1.107
1.089
1.075
1.065
1.056
,044
.036
.029
,024
.020
.015
1.
01 1
1.007
1.003
1.001
1 .Ooo
1.713
1.359
1.238
1.175
1.136
1.110
1.09
1 1.
077
1.066
1.058
1.045
1.036
1.029
I .024
1.021
1.01
6 1.01 1
1.007
I .003
1.001
1 .#
85.5271
90.5312 95.0231 100.4250 104.2150
1-
2
1.29
7 3
1.20
7 4
1.15
7 5
1.12
5
6 1.
102
7 1.
086
8 1.
073
9 1.
063
10
1.05
5
12
1.04
3 14
1.
035
16
1.02
9 18
1.
024
20
1.02
1
24
1.01
6 30
1.
011
40
1.00
7 60
1.00
3 I2
0 1.
001
m
1.O
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xz,,,
93.2
70
-
1.31
5 1.
218
1.16
4 1.
130
1.10
6
Table9
(Contin
ued
)
m=
7 -
r=ll
r=
12
a
0.10
0 0.050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.025
0.10
0 0.
005
1.62
7 1.
690
1.73
7
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76
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57
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36
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1.
025
1.02
1
1.01
6 1.
01 1
1.00
7 1.
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1.00
1
1 .Ooo
98.4
84
-
1.33
2 I .
227
1.17
1 1.
135
1.1
10
1.09
2 1.
078
1.06
7 I .0
59
1.04
6 I .
037
I .03 1
1.
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1.02
2
1.01
7 1.
012
1.00
7 1.
004
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1 .OOo
10
3.1 5
8
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1.354
1.23
9 1.
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1.14
0
1.11
4 1.
095
1.08
1 1.
070
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1
1.04
8 1.
038
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027
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3
1.01
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1
1 .OOo
108.
771
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1.37
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1.18
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1.06
2
1.04
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039
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1.01
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1 .Ooo
11
2.70
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1.53
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I .No8
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1.57
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1.22
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8
1.11
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070
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2
1.04
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1.01
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1.00
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1
1 .O
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I .02
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I .34
4 1.
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1.18
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1.11
7 1.
098
1 .OM
1.07
3 I .064
1 .ow
1.04
1 1 .OM
1.02
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024
1.01
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I .Ooo
1.36
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w)
1.18
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149
1.12
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02
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1.02
5
1.01
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1 .Ooo
1.38
3 1.
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1.19
4 1.
153
1.12
5 1.
104
I .089
1.
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1.01
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1 .Ooo
100.
9800
106
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1.24
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17.0
565
121.
1263
r=1
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r=1
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0.10
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0.05
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0.01
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2
1.32
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1.22
8 4
1.17
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1.14
1
6 1.
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7 1.
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8 1.
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9 1.
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10
1.06
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12
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1 14
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24
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1.35
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1.71
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1.09
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r=1
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1.21
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r=1
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-
r =8
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85
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53
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023
1.02
0
1.01
5 1.
01 1
1.00
6 1.
003
1.00
1
1 .Ooo
1.54
7 1.
303
1.20
9 1.
158
1.12
5
I. 10
2 1.
086
1.073
1.
063
1.05
5
1.04
3 1.
035
1.02
9 1.
024
1.02
1
1.10
6 1.
01 1
1.00
7 I .
003
1.00
1
1 .Ooo
1.59
3 1.
319
1.21
9 1.
164
1.13
0
1.10
6 1.
088
I .075
1.065
1.
057
1.04
4 1.
036
1.03
0 1.
025
1.02
1
1.10
6 1.
01 1
1.
007
1.00
3 1.
001
1 .Ooo
1.65
3 1.
341
1.23
0 1.
172
1.13
5
1.1
10
1.09
2 1.
078
1.06
7 1.
059
1.04
6 1.
037
1.03
0 1.
026
1.02
2
1.01
6 1.
01 I
1.00
7 1.
003
1.00
1
1 .m
1.69
8 1.
357
I .239
1.
177
1.13
9
1.11
3 1.
094
1.08
0 1.
069
1.06
0
1.04
7 1.
038
1.03
1 1.
026
1.02
2
1.01
7 1.
012
1.00
7 1.
004
.I .00
1
1 .Ooo
%.5
782
101.
8795
10
6.62
86
112.
3288
11
6.32
1 1
r=ll
r=12
0.050
0.025
0.010
0.005
0.100
0.050
0.025
0.010
0.005
1-
2
1.294
3 1.208
4 1.159
5 1.127
6 1.104
7 1.088
8 1.075
9 1.065
10 1.057
12
1.045
14
1.037
16
1.030
18
1.026
20
1.022
24
1.017
30
1.012
40
1.007
60
1.004
120
1.001
00
1.OOo
-
1.312
1.218
1.165
1.132
I. 108
1.09
1 1.078
I .067
1.059
I .046
1.038
1.03 1
I .026
1.022
1.017
1.012
I .007
1.00
4 1.001
1 .Ooo
-
1.328
1.227
1.172
1.136
1.112
I .094
1.080
1.069
1.061
1.048
1.039
1.032
1.027
1.023
1.017
1.012
1.008
1.00
4 1.001
1 .Ooo
-
1.350
1.239
1.179
I. 142
1.1 I6
1.097
1.083
I .072
1.063
I .049
I .040
1.033
I .028
1.024
1,018
1.013
1.008
1.00
4 1.001
1 .OOo
-
1.366
1.247
1.185
1.14
6
1.119
1.100
I .085
1.073
1.06
4
1.050
1.041
1.034
I .028
I .024
1.01
8 1.013
1.008
I .004
I .
001
1 .Ooo
x;,,,
105.372
110.898
I15.841
121.767
125.913
1.516
1.30
4 1.216
1.166
1.134
1.11
1 1.093
1.080
1.070
1.061
,049
.039
.033
.028
.024
.018
1.013
1.008
1.00
4 1.001
1 .O
oo
1.562
1.321
1.226
.I73
.I39
,114
.097
.083
.072
.063
1.050
1.041
1.034
1.029
1.024
1.019
1.013
I .008
1.00
4 1.001
1 .OOo
1.608
I .338
1.236
1.180
1.143
1.1 I8
I .ow
1.085
I .074
1.065
1.05 1
1 .042
1.035
I .029
1.025
1.019
1.013
1.008
1.00
4 1.
001
1 .Ooo
1.667
I .359
1.248
1.187
1.149
1.122
1.103
I .088
1.076
1.067
1.053
1.043
I .036
1.030
1.026
1.020
1.014
1.009
1.00
4 1.001
1 .Ooo
1.713
1.375
1.256
1.193
1.153
1.126
1.105
1.09
0 1.078
1.068
.OH
.0
44
.036
.03 1
.026
.020
1.01
4 1.009
1.00
4 1.001
1 .Ooo
114.1307 119.8709 125.0001 131.1412 135.4330
Tab
le 9
(C
on
fke
d)
r=1
3
r= 1
4 a
0.10
0 0.050
0.02
5 0.
010
0.00
5 0.
100
0.050
0.02
5 0.
010
0.00
5
l-
2
1.31
3 3
1.22
5 4
1.17
4 5
1.14
1
6 1.
117
7 1.
099
8 1.
085
9 1.
074
10
1.06
6
12
1.05
2 14
1.
043
16
1.03
5 18
1.
030
20
1.02
6
24
1.02
0 30
1.
014
40
1.009
60
1.00
4 12
0 1.
001
00
1.OOo
x;,
122.
858
-
1.33
1 1.
235
1.18
1 1.
146
1.12
1 1.
102
1.08
8 1.
077
1.06
7
1.05
4 1.
044
I .036
I .
03 1
1.02
7
I .020
1.
014
1.00
9 1.
004
1.00
1
I .Ooo
28.8
04
-
1.34
7 1.
245
1.18
8 1.
151
1.12
5 1.
105
1.09
0 1.
079
1.06
9
1.05
5 1.
045
1.03
7 1.
032
1.02
7
1.02
1 1.
015
1.00
9 1.
005
1 .oOl
1 .Ooo
34.1
11
~ .
-
1.36
9 1.
257
1.1%
1.
156
1.12
9 1.
109
1.09
3 1.
08 1
1.07
I
1.05
7 1.
046
1.03
8 1.
033
1.02
8
1.02
I 1.
015
1.00
9 1.
005
1.00
1
1 .Ooo
14
0.45
9
-
1.38
5 1.
266
1.20
1 1.
161
1.13
2 1.
111
1.09
6 I .
083
1.07
3
1.05
8 1.
047
1.03
9 1.
033
1.02
8
1.02
2 1.
015
1.01
0 1.
005
1.00
1
1 .Ooo
44.8
91
1.53
5 1.
323
1.23
4 1.
182
1.14
8
1.12
3 1.
105
1.09
1 1.
079
t ,07
0
1.05
6 1.
046
1.03
8 1.
032
1.02
8
1.02
1 1.
015
1.01
0 1.
005
1.00
1
1 .ooo
1.58
1 1.
626
1.34
1 1.
357
1.24
4 1.
254
1.18
9 1.
1%
1.15
3 1.
158
1.12
7 1.
131
1.10
8 1.
111
1.09
3 1.
0%
1.08
2 1.
084
1.07
2 1.
074
1.05
7 1.
059
1.04
7 1.
048
1.03
9 1.W
1.
033
1.03
4 1.
029
1.02
9
1.02
2 1.
022
1.01
6 1.
016
1.01
0 1.
010
1.00
5 1.
005
1.00
1 1.
001
1.OOo
1.O
Oo
1.68
6 1.
379
1.26
6 1.
204
1.16
4
1.13
6 1.
1 15
1.
099
1.08
6 1.
076
1.06
1 1.
050
1.04
1 1.
035
1.03
0
I .023
1.
016
1.01
0 1.
005
1.00
1
1 .Ooo
1.73
1 1.
395
1.27
5 1.
210
1.16
8
1.13
9 1.
1 18
1.
101
1.08
8 1.
078
1.06
2 1.
051
1.04
2 1.
036
1.03
1
1.02
3 1.
017
1.01
0 1.
005
1.00
1
1 .OO
o
31.5
576
137.
7015
143
.180
1 14
9.72
69 1
54.2
944
r=15
r=
16
u
0.
100
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
I-
2
1.33
3 3
1.24
3 4
1.19
0 5
1.15
5
6 1.
130
7 1.
111
8 1.
0%
9 1.
084
10
1.07
4
12
1.06
0 14
1.
049
16
1.04
1 18
1.
035
20
1.03
0
24
1.02
3 30
1.
016
40
1.01
0 60
1.00
5 12
0 1.
002
00
1.OOo
-
1.35
1 I .
253
1.19
8 1.
160
1.13
4 1.
114
1 .O
N
1.08
7 1.
076
1.06
1 1.
050
1.04
2 1.
036
1.03
1
1.02
4 1.
017
1.01
1 1.
005
I .00
2
1 .Ooo
-
1.36
8 1.
263
1.20
4 1.
165
1.13
8 1.
117
1.10
1 1.
089
1.07
8
1.06
3 I .0
52
1.04
3 1.
037
1.03
2
1.02
4 1.
017
1.01
I 1.
005
1.00
2
I .Ooo
-
1.38
9 1.
275
1.21
2 1.
171
1.14
3 1.
121
1.10
5 1.
091
I .08
1
I .06
5 1.
053
1.04
4 1.
038
1.03
2
1.02
5 1.
018
1.01
I I .
006
I .002
1 .Ooo
-
1.40
6 1.
284
1.21
8 1.
176
1.14
6 1.
124
1.10
7 I .0
93
1.08
2
1.06
6 I .054
1.04
5 1.
038
1.03
3
1.02
5 1.
018
1.01
1 I .006
1.00
2
1 .Ooo
xf
, 14
0.23
3 14
6.56
7 15
2.2l
l 15
8.95
0 16
3.64
8
1.55
5 I .3
43
1.25
2 1.
198
1.16
2
1.13
6 1.
117
1.10
1 1.
089
1.07
9
1.06
4 1.
052
1.04
4 1.
038
1.03
2
1.02
5 1.
018
1.01
1 1.
006
I .002
I .Ooo
1.60
1 1.
36 1
1.26
3 1.
206
1.16
8
1.14
1 1.
120
1.10
4 I .0
92
1.08
1
1.06
5 I .0
54
1.04
5 1.
038
1.03
3
1.02
6 1.
018
1.01
2 1.
006
I .002
1 .Ooo
I .646
1.37
8 I .2
72
1.21
2 1.
173
1.14
5 1.
123
1.10
7 1.
094
1.08
3
1.06
7 1.
055
1.04
6 1.
039
I .034
I .026
1.
019
1.01
2 I .006
I .002
1 .Ooo
I .706
1.
400
1.28
5 1.
221
1.17
9
1.14
9 1.
127
1.1 1
0 1.
097
1.08
5
1.06
9 I .
056
1.04
7 1.
040
I .035
1.02
7 1.
019
1.01
2 1.
006
I .002
I .Ooo
I .75
1 1.
416
I .29
4 ,2
21
. I83
.153
.1
30
.113
,099
.087
1.07
0 1.
057
I .04
8 1.
041
1.03
5
1.02
7 1.
019
1.01
2 1.
006
I .002
1 .Ooo
148.
8853
155
.404
7 16
1.20
87 1
68.1
332
172.
9575
m P
Tab
le9
(Conllr
ared
)
m=
8 -
r=1
7
r=1
8
(I
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1.57
5 1.
621
1.66
7 1.
727
1.36
3 1.
381
1.27
0 1.
281
1.21
5 1.
222
1.17
7 1.
183
1.15
0 1.
154
1.12
9 1.
133
1.11
2 1.
115
1.09
9 1.
102
1.08
8 1.
091
1.07
1 1.
073
1.05
9 1.
061
1.05
0 1.
051
1.04
3 1.
044
1.03
7 1.
038
1.02
9 1,
029
1.02
1 1.
021
1.01
3 1.
014
1.00
7 1.
007
1.00
2 1.
002
1.OOo
1.
Ooo
1.39
8 1.
291
1.22
9 1.
188
1.15
8 1.
136
1.11
8 1.
104
1.09
3
1.07
5 1.
062
I .05
2 1.
045
1.03
9
1.03
0 1 .O
22
1.01
4 I .
007
1.00
2
1 .Ooo
1.42
0 1.
304
1.23
8 1.
194
1.16
3 1.
140
1.12
2 1.
107
1.09
5
1.07
7 1.
064
1.05
4 1.
046
1.04
0
1.03
1 1.
022
1.01
4 1.
007
1.00
2
1 .Ooo
1 2 3 4 5 6 7 8 9 10
12
14
16
18
20
24
30
40
60
120
a2
-
1.35
3 1.
261
1.20
7 1.
170
1.14
3 1.
123
1.10
7 I .094
1.08
4
1.06
7 1.
056
1.04
7 1.
040
1.03
5
1.02
7 1.
019
1.01
2 1.
006
1.00
2
1 .Ooo
-
1.37
1 1.
272
1.21
4 1.
175
1.14
7 1.
126
1.11
0 1.
097
1.08
6
1.06
9 1.
057
1.04
8 1.
041
I .036
1.02
7 1.
020
1.01
3 1.
006
1.00
2
1 .Ooo
-
1.38
8 1.
282
1.22
1 1.
180
1.15
1 1.
130
1.1 1
3 1.
099
I .088
1.07
1 1.
058
1.04
9 1.
042
1.03
6
1 .OX8
1.02
0 1.
013
I .00
7 1.
002
1 .O
oo
-
-
1.41
0 -
1.29
4 1.
303
1.22
9 1.
235
1.18
7 1.
191
1.15
6 1.
160
1.13
4 1.
136
1.11
6 1.
118
1.10
2 1.
104
1.09
0 1.
092
1.07
3 1.
074
1.06
0 1.
061
1.05
0 1.
051
1.04
3 1.
044
1.03
7 1.
038
1.02
9 1.
029
1.02
1 1.
021
1.01
3 1.
013
1.00
7 1.
007
1.00
2 1.
002
1.OOo
1.
Ooo
1.77
3 1.
437
1.31
3 1.
244
1.19
9
1.15
7 1.
143
1.12
4 1.
109
1.09
7
I .07
8 1.
065
1.05
5 1.
047
1 .040
1.03
1 1.
022
1.01
4 I .
007
1.00
2
1 .Ooo
x;
, 15
7.51
8 16
4.21
6 17
0.17
5 17
7.28
0 18
2.22
6 16
6.13
18 1
73.0
041
179.
1 I37
18
6.39
30 1
91.4
585
m=
9 -
r=9
r=
lO
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1 1.
495
2 1.
282
3 1.
197
4 1.
149
5 1.
119
6 1.
097
7 1.
081
8 1.
069
9 1.
060
10
1.05
2
12
1.04
1 14
1.
033
16
1.02
7 18
1.
023
20
1.02
0
24
1.01
5 30
1.
010
40
1.00
6 60
1.
003
120
1.00
1
m
1.w
I .54
0 1.
299
1.20
7 1.
156
1.12
3
1.10
1 I .OM
I .07
2 I .
062
1.05
4
1.04
3 I .
034
1.02
8 1.
024
1.02
0
1.01
5 1.
011
1.00
7 1.
003
1 .00
1
1 .O
oo
1.58
5 1.
315
1.21
6 1.
162
1.12
8
1.10
4 1.
087
1.07
4 I .064
1.05
6
1.04
4 1.
035
1.02
9 1.
024
1.02
1
1.01
6 1.
01 1
I .007
1.
003
1.00
1
1 .Ooo
1.64
5 1.
337
1.22
7 1.
169
1.13
3
1.10
8 I .ow
1.07
7 1.
066
I .05
8
I .045
1.
036
1.03
0 1.
025
1.02
2
1.01
6 1.
01 1
1.00
7 1.
003
1.00
1
1 .Ooo
1.69
0 I .3
53
1.23
6 1.
175
1.13
7
1.1 1
1 1.
093
1.07
9 1.
068
1.05
9
1.04
6 1.
037
1.03
I 1.
026
1.02
2
1.01
7 1.
012
I .007
1.
003
I .001
1 .Ooo
x:,
97.6
7% 1
03.0
095
107.
7834
113
.512
4 11
7.52
42
1.49
7 I .
288
I .203
1.
155
1.12
4
1.10
2 1.
086
1.07
3 I .0
64
1.05
6
1.04
4 I .0
36
1.03
0 1.
025
1.02
I
1.01
6 1.
01 1
I .007
1.
003
1.00
1
1 .Ooo
1.54
2 1.
305
1.21
3 1.
162
1.12
9
1.10
6 1.
089
1.07
6 1.
066
I .058
1.04
5 1.
037
1.03
0 1.
026
I .02
2
1.01
7 1.
012
1.00
7 1.
004
1.00
1
1 .OOO
1.58
6 1.
321
I .222
1.
168
1.13
3
1. I0
9 1.
092
1.07
8 I .0
68
I .05
9
1.04
7 1.
038
1.03
1 1.
026
I .023
1.01
7 1.
012
1.00
7 1 .w
1.
001
I .Ooo
I .645
1.
342
1.23
3 1.
175
1.13
9
1.1
I3
1.09
5 1.
08 1
1.07
0 1.
061
1.04
8 1.
039
I .032
1.
027
I .023
1.01
8 1.
012
I .008
I .m
1.
001
1 .Ooo
1.69
0 1.
357
1.24
2 1.
181
1.14
3
1.11
6 1.
097
1.08
3 1.
072
1.06
3
1.04
9 I .0
40
1.03
3 1.
028
1.02
4
1.01
8 1.
013
1.00
8 I .0
04
1.00
1
1 .Ooo
10
7.56
50 1
13.1
453
118.
1359
124
.1 1
63 1
28.2
989
QI e
Tab
te9
(Contin
ued
)
m =
9
r=
ll
r=1
2
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
1-
2
1.29
4 3
1.21
0 4
1.16
1 5
1.13
0
6 1.
107
7 1.
091
8 1.
078
9 1.
068
10
1.05
9
12
1.04
7 14
1.
038
16
1.03
2 18
1.
027
20
1.02
3
24
1.01
8 30
1.
012
40
1.00
8 60
1.
004
120
1.00
1
03
1.OOo
x;,
1 17.
407
-
1.31
1 1.
219
1.16
8 1.
134
1.11
1 1.
094
1.08
0 1.
070
1.06
1
I .04
8 1.
039
1.03
3 1.
028
1.02
4
1.01
8 1.
013
1.00
8 1.
004
1.00
1
1 .ooo
123.
225
-
1.32
7 1.
229
1.17
4 1.
139
1.1 1
4 I .0
96
1.08
3 1.
072
1.06
3
I .050
1.
040
1.03
4 1.
028
1.02
4
1.01
8 1.
013
1.00
8 1.
004
1.00
1
1 .OoO
128.
422
-
I .348
1 2
40
1.18
2 1.
144
1.11
9 1.
100
1.08
5 1.
074
I .065
1.05
I 1.
042
1.03
5 1.
029
1.02
5
1.01
9 1.
013
1.00
8 1.
004
1.00
1
1 .Ooo
134.
642
-
I .364
1.
248
1.18
7 1.
148
1.12
2 1.
102
1.08
7 1.
076
1.06
6
1.05
2 1.
043
1.03
5 1.
030
1.02
5
1.01
9 1.
014
1.00
8 1.
004
1.00
1
1 .Ooo
138.
987
I so6
1.
302
1.21
7 1.
168
1.13
6
1.11
3 1.
095
I .082
1.
072
1.06
3
1.05
0 1.
041
1.03
4 1.
029
1.02
5
1.01
9 1.
013
1.00
8 I .0
04
1.00
1
I .Ooo
.550
.3
19
.227
.1
75
.I41
.I16
1.
099
I .08
5 1.
074
1.06
5
1.05
2 1.
042
1.03
5 1.
030
1.02
6
1.01
9 1.
014
1.00
9 1.
004
1.00
1
1 .Ooo
1.594
I .
335
1.23
6 1.
181
1.14
5
1.12
0 1.
101
1.08
7 1.
076
1.06
7
I .053
1 .@
I3
1.03
6 1.
030
1.02
6
1.02
0 1.
014
1.00
9 1.
004
1.00
1
1 .OOo
1.45
2 1.
355
1.24
7 1.
188
1.15
1
1.12
4 1.
105
I .ow
1.
078
1.06
9
1.05
5 1.
044
1.03
7 1.
03 1
1.02
7
1.02
0 1.
014
1.00
9 1.
005
1.00
1
1 .Ooo
~
1.6%
1.
37 1
1.
256
1.19
4 1.
155
1.12
7 1.
107
1.09
2 1.
080
1.07
0
I .056
1.
045
1.03
8 1.
032
1.02
7
1.02
1
1.01
5 1.
009
1 .OM
1.00
1
1 .Ooo
127.
21 1 I
13
3.25
69 1
38.6
506
145.
0988
149
.599
4
r=1
3
r=1
4
(1
0.10
0 0.
050
0.025
0.
010
0.00
5 0
.10
0.
050
0.02
5 0.
010
0.00
5
1-
2
1.31
0 3
1.22
4 4
1.17
5 5
1.14
2
6 1.
118
7 1.
101
8 1.
087
9 1.
076
10
1.06
7
12
14
16
18
20
24
,054
.w
,0
37
.03
I .0
27
,020
30
1.
015
40
1.00
9 60
1.00
5 12
0 1.
001
w
1.O
oo
-
I .326
1.
234
1.18
1 I.
I47
1.12
2 1.
104
I .089
1.
078
1.06
9
.055
,045
.038
.0
32
.028
.02 1
1.
015
1.00
9 1.
005
1.00
1
1 .Ooo
-
1.34
3 1.
243
1.18
8 1.
151
1.126
1.
107
I .092
I .0
80
1.07
1
1.05
6 1.
046
I .039
1.0
33
1.028
1.02
1 1.
015
1.01
0 I .0
05
1.00
1
1 .Ooo
-
1.36
3 1.
255
1.19
5 1.
157
1.130
1.1
10
I .095
I .0
83
1.07
3
,058
.0
47
.QQo
.0
34
.029
.022
1.
016
1 .Ol
O I .0
05
I .00
1
1 .Ooo
-
1.379
1.2
63
1.20
1 1.
161
1.13
3 1.
113
1.09
7 I .0
84
1.074
1.05
9 1.
048
I .040
1.03
4 I .0
29
I .022
1.
016
1.01
0 I .
005
1.00
1
I .Ooo
I .520
1.
318
1.23
2 1.
182
1.14
8
1.12
4 1.
106
1.09
2 1.
080
1.07
1
1.05
7 1.
047
1.03
9 1.
033
1.02
9
1.02
2 1.
016
1.01
0 1.
005
1.00
1
I .Ooo
1.56
3 I .3
35
I .242
1.
189
1.15
1
1.12
8 1.
109
1.09
4 1.
083
1.07
3
1.05
9 1.
048
I .w
1.03
4 I .0
30
1.02
3 1.
016
1.01
0 1.
005
1.00
1
I .Ooo
xs,,
136.
982
143.
246
148.
829
155.
4%
160.
146
I .607
I .3
5 1
1.25
I 1.
195
1.15
8
1.13
2 1.
112
1.097
1.08
5 1.0
75
1.06
0 1.
049
1.04
1 1.
035
1.03
0
I .02
3 1.
016
1.01
0 I .005
1.00
2
1 .Ooo
I .664
1.37
1 1.
263
1.20
3 1.
164
1.13
6 1.
116
1.10
0 1.
087
1.07
7
1.06
2 1.
05 1
I .04
2 1.
036
1.03
1
1.02
4 1.
017
1.01
I 1.
005
1.00
2
1 .Ooo
I .70
8 1.
387
1.27
1 1.
208
1.16
8
1.13
9 1.
118
1.10
2 I .
089
1.07
9
1.06
3 1.
051
I .04
3 1.
037
I .03
2
1.02
4 1.
017
1.01
1 1.
005
1.00
2
I .OOo
146.
7241
15
3.19
79 1
58.%
24
165.
8410
170
.634
1
1-
-
2 1.
326
1.34
3 3
1.24
0 1.
250
4 1.
189
1.1%
5
1.15
5 1.
160
6 1.
130
1.13
4 7
1.11
1 1.
115
8 1.09
7 1.
099
9 1.
085
1.08
7 10
1.
075
1.07
7
12
1.06
1 1.
062
14
1.05
0 1.
051
16
1.04
2 1.
043
18
1.03
6 1.
037
20
1.03
1 1.
032
24
1.02
4 1.
024
30
1.01
7 1.
017
40
1.01
1 1.
011
60
1.00
5 1.
006
120
1.00
2 1.
002
00
1.OOo
1.O
Oo
-
-
-
1.35
9 -
-
1.25
9 1.
271
1.27
9 1.
202
1.21
0 1.
216
1.16
5 1.
170
1.17
4
1.138
1.
142
1.14
5 1.
118
1.12
1 1.
124
1.10
2 1.
105
1.10
7 1.
089
1.09
2 1.
094
1.07
9 1.
081
1.08
3
1.06
4 1.
065
1.06
6 1.
052
1.05
4 1.
055
1.04
4 1.
045
1.04
6 1.
037
1.03
8 1.
039
1.03
2 1.
033
1.03
4
1.02
5 1.
025
1.02
6 1.
018
1.01
8 1.
018
1.01
1 1.
012
1.01
2 1.
006
1.00
6 1.
006
1.00
2 1.
002
1.00
2
1.OOo
1.O
Oo
1.OOo
Tab
le 9
(C
mrh
ued
)
m=
9
-
r=15
r=
16
a
0.10
0 0.
050
0.02
5 0.
010
0.00
5 0.
100
0.05
0 0.
025
0.01
0 0.
005
1.53
6 1.
579
1.62
2 1.
679
1.72
2 1.
335
1.35
2 1.
248
1.25
8 1.
1%
1.20
3 1.
161
1.16
6
1.13
6 1.
140
1.11
7 1.
120
1.10
2 1.
104
1.08
9 1.
092
1.07
9 1.
082
1.06
4 1.
066
1.053
1.
054
1.04
5 1.
046
1.03
8 1.
039
1.03
3 1.
034
1.02
6 1.
026
1.01
8 1.
019
1.01
2 1.
012
1.00
6 1.
006
1.00
2 1.
002
1.oo
o 1.O
Oo
1.36
8 1.
267
1.21
0 1.
171
1.14
4 1.1
23
1.107
1.
094
1.08
3
1.06
7 1.
056
1.04
7 1.
040
1.03
5
1.02
7 1.
019
1.01
2 1.
006
1.00
2
1 .Ooo
1.38
9 1.
279
1.21
8 1.
177
1.14
8 1.
127
1.1
10
1.09
7 1.
086
1.06
9 1.
057
1.04
8 1.
041
1.03
5
1.02
7 1.
020
1.01
2 1.
006
1.00
2
1 .OOo
1.40
4 1.
288
I .223
1.
181
1.15
2 1.
130
1.1
12
I .099
1.08
7
1.07
0 1.
058
1.04
9 1.
042
1.03
6
1.02
8 1.
020
1.013
1.
006
1.00
2
1 .OOo
xs
, 15
6.44
0 16
3.1
16
169.
056
176.
138
181.
070
166.
1318
173
.004
1 17
9.11
37
186.
3930
191
.458
5
m=
lO
r=lO
r=
ll
0.05
0 0.
025
0.01
0 0.
005
0.10
0 0.
050
0.02
5 0.
010
0.00
5
I 1.
4%
2 1.2
91
3 1.
208
4 1.
160
5 1.
128
6 1.
106
7 1.
090
8 1.
077
9 1.
067
10
1.05
9
12
1.04
7 14
1.
038
16
1.03
1 18
1.
027
20
1.02
3
24
1.01
7 30
1.
012
40
1.00
8 60
1.00
4 12
0 1.
001
M
1.Ooo
I .54
0 I .3
08
1.21
7 1.
166
1.13
3
1.11
0 I .0
93
1.07
9 1.
069
1.06
1
1.04
8 I .0
39
1.03
2 1.
027
1.02
3
1.01
8 1.
01 3
1.00
8 1.
004
1.00
1
I .Ooo
1.58
4 1.
324
1.2.26
1.
172
1.13
7
1.11
3 1.
095
I .082
1.
07 1
1.06
2
1.04
9 1 .oQo
1.03
3 1.
028
1.02
4
1.01
8 1.
013
I .008
1 .m
1.00
1
I .Ooo
1.64
1 1.
345
1.23
8 1.
180
1.14
3
1.11
7 1 .o
w
I .OM
1.07
3 1.
064
1.05
1 1.
041
1.03
4 1.
029
1.02
5
1.01
9 1.
01 3
1.00
8 1.
004
1.00
1
1 .Ooo
1.58
6 I .
360
1.24
6 1.
185
1.14
7
1.12
0 1.
101
I .08
6 1.
075
1.06
6
1.05
2 1.
042
1.03
5 1.
029
1.02
5
1.01
9 1.
013
1.00
8 1.
004
1.00
1
1 .Ooo
x:,,,
18.4
980
124.
3421
129
.561
2 13
5.80
67 1
40.1
695
-
1.2%
1.
213
1.16
5 1.
133
1.11
1 1.
094
1.08
I I .
070
1.06
2
I .04
9 1.
040
1.03
4 1.
028
I .02
4
1.01
9 1.
013
I .008
1.00
4 1.
001
1 .Ooo
12
9.38
5
-
1.31
3 1.
222
1.17
1 1.
138
1.1
14
1.09
7 1.
083
1.07
2 1.
064
I .05
1 1-
041
1.03
4 1.
029
1.02
5
1.01
9 I .
013
I .00
8 1.
004
1 .00
1
1 .Ooo
135.
480
-
I .32
9 1.
231
1.17
7 1.
I42
1.1
18
1.09
9 1.
085
1.07
4 1.
065
1.05
2 I .0
42
1.03
5 1.
030
1.02
6
I .02
0 1.
014
1.00
9 1.
004
1.00
1
1 .Ooo
14
0.91
7
-
-
1.34
9 -
1.24
3 1.
251
1.18
5 1.
190
1.14
8 1.
152
1.12
2 1.
125
1.10
3 1.
105
1.08
8 1.
090
1.07
7 1.
078
1.06
7 1.
069
1.05
4 1.
055
1.04
4 1.
044
1.03
6 1.
037
1.03
1 1.
031
1.02
6 1.
027
1.02
0 1.
020
1.01
4 1.
014
1.00
9 1.
009
1.00
4 1.
005
1.00
1 1.
001
1.OOo
1.
Ooo
147.
414
151.
948
Tab
le 9
(C
onri
nu
ed)
~ ~
~~
~
m=
lO
r=1
2
r=14
a
0.10
0 0.
050
0.02
5 0.
010
@.0
05
0.10
0 0.050
0.02
5 0.
010
3.005
I 1.
500
1.54
3 1.
585
1.64
1 1.
684
1.50
9 1.
551
1.59
3 1.
648
1.69
0 2
1.30
2 1.
318
1.33
4 1.
354
3 1.
219
1.22
8 1.
237
1.24
8 4
1.17
0 1.
177
1.18
3 1.
190
5 1.
138
1.14
3 1.
148
1.15
3
6 1.
115
1.11
9 1.
123
1.12
7 7
1.09
8 1.
101
1.10
4 1.
107
8 1.
085
1.08
7 1.
090
1.09
2 9
1.07
4 1.
076
1.07
8 1.
081
10
1.06
5 1.
067
1.06
9 1.
071
12
1.05
2 1.
054
1.05
5 1.
057
14
1.04
3 1.
044
1.04
5 1.
046
16
1.03
6 1.
037
1.03
8 1.
039
I8
1.03
0 1.
031
1.03
2 1.
033
20
1.02
6 1.
027
1.02
7 1.
028
24
1.02
0 1.
020
1.02
1 1.
021
30
1.01
4 1.
015
1.01
5 1.
015
40
1.00
9 1.
009
1.00
9 1.
010
60
1.00
4 1.
005
1.00
5 1.
005
120
1.00
1 1.
001
1.00
1 1.
001
30
1.OOo
1.O
Oo
1.OOo
1.O
Oo
xf,
140.
2326
14
6.56
74
152.
21 1
4 15
8.95
02
1.36
9 1.
257
1.1%
1.
157
1.13
0 1.
110
1.09
4 1.
082
1.07
2
1.05
8 1.
047
I .03
9 1.
033
1.02
9
1.02
2 1.
015
1.01
0 1.
005
1.00
1
1 .OoO
163.
6482
1.31
5 I .
232
1.18
2 1.
149
1.12
6 1.
107
1.09
3 1.
082
1.07
3
I .058
1.
048
i .040
1.03
5 1.
030
1.02
3 1.
016
1.01
0 1.
005
I .00
2
1 .O
oo
1.33
1 1.
241
1.18
9 1.
154
1.12
9 1.
111
1.09
6 1 .OM
1.07
5
1.06
0 1.
049
1.04
2 1.
035
I .03
1
1.02
4 1.
017
1.01
1
1.00
5 1.
002
1 .ooo
1.34
7 1.
250
1.19
5 1.
159
1.13
3 1.
113
1.09
8 1.
086
1.07
6
1.06
1 1.
05 1
I .
042
1.03
6 1.0
3 1
1.02
4 1.
017
1.01
1
1.00
6 1.
002
I .OOo
1.36
7 1.
382
1.26
1 1.
269
1.20
3 1.
208
1.16
4 1.
168
1.13
7 1.
141
1.11
7 1.
119
1.10
1 1.1
03
1.08
9 1.
090
1.07
8 1.
080
1.06
3 1.
064
1.05
2 1.
053
1.04
3 1.
044
1.03
7 1.
038
1.03
2 1.
033
1.02
5 1.0
25
1.01
8 1.
018
1.01
1 1.
011
1.00
6 1.
006
1.00
2 1.
002
1.OOo
1.O
Oo
161.
8270
16
8.61
30 1
74.6
478
181.
8403
186.
8468
m=12
r=12
a
0.10
0 0.
050
0.025
0.010
0.005
1 2 3 4 5 6 7 8 9
10
12
14
16
18
20
24
30
40
60
120 00
2 X
rm
1.495
1.30
6 1.225
1.145
1.122
1.104
1.091
1.080
1.07 1
1.057
I .047
1.039
1.034
1.029
1.022
1.016
1.010
1.005
1.001
1 .ooo
1.m
1.322
1.234
1.184
1.15
0
1.126
1.107
1.093
1.082
1.072
1.058
1.04
8 1.
040
1.034
1.030
1.023
1.016
1.010
1.005
1.002
1 .Ooo
1.576
1.337
1.243
1.190
1.154
1.129
1.110
1.095
1.084
1.074
1.060
1.049
1.041
1.035
1.030
1.023
1.017
1.01 1
1.005
1.002
1 .Ooo
1.630
1.356
1.25
4 1.197
1.16
0
1.133
1.1 14
I .09
8 1.086
1.076
1.061
1.05
0 1.042
1.036
1.031
1.024
1.017
1.01
1 1.005
1.00
2
I .OOo
1.67 1
1.371
1262
1.20
2 1.163
1.136
1.116
1.100
1.088
1.078
1.062
1.05 1
1.043
1.037
1.032
1.024
1.017
1.01 1
1.
006
1.002
1 .OOo
166.1318 173.0041 179.1137 186.3930 191.4585