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n retail businesses, factories, and of ces such as IBM, Safeway, General Electric, and Daimler Chrysler, education or training and development, as it is called in the private sector is a staple activity The pace of innovation and adaptation in technical, manufacturing, and service organizations has been phenomenal Robots, computers, lasers, and other developments in technology, including E-learning (learning via the Internet and other electronic means), have brought about a critical need to retrain technical employees Constantly changing, more sophisticated approaches to marketing have necessitated the upgrading of personnel Experimentation and ndings in personnel management, training, and ways to promote employee growth have created a wholly new eld: human resource development Other events that have given impetus to education in the private sector are foreign competition and the emergence of the employee as a company partner The rising industrial prowess of foreign countries, many with much cheaper labor costs, has devastated or emasculated some basic industries in the United States, brought about the ring of many in the workforce, and created the need to retrain thousands of employees for new jobs The realization that employees need cultivation, involvement, nurturing, and support to function well and productively has redirected the focus of the business and industrial training effort from a concentration on skills alone to a broad concern with employee development Training new employees, upgrading veteran workers, and introducing new techniques, technologies, and tactics all that amounts to big busi177

Third xl r flxo, xt,x2,

SELECT DATEADD(mm,-number,CONVERT(varchar(8),GETDATE(),112)) FROM dbo.Numbers WHERE number < 100 ORDER BY number

FIGURE 5 I5 Grophicol depiction iherecursive of finite is of noture divided This differences representotion ,eferred oso divided to difference toble

2008-10-29 2008-09-29 2008-08-29 .... .... 2000-07-29

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(1514), Thesedifferences be usedto evaluate coefficients Eqs(1511) can the in through which can then be substituted interinto Eq (1510)to yield the generalform of Newton's polating polynomial: f,-rG):f ( x t ) + ( r - x t ) f l x z r r l * ( " r- x r ) ( x - x ) f l 4 , x 2 , x 1 l + " + ( x - x r ) ( x - r z ) ( r - x n - t ) f l x n , t r n - t , , x z , x r l( 1 5 1 8 )

We should note that it is not necessary that the data points used in Eq (1518) be valuesnecessarily in ascending equally spacedor that the abscissa be order,asillustrated in the following exarnple Also noticehow Eqs (1515)through(1517)arerecursivethat is, higher-order differences are computed by taking differences of lower-order differences(Fig 155)This property will be exploitedwhen we developan efficientM-file in Section1524 implement method to the 1 EXAMPLE 54 Newton Interpoloting olynomiol P Problem Siotement In Example 153,data pointsat xt : l, xz:4, and,r3= ( 1ry91s = used to estimate ln 2 with a parabola Now, adding a fourth point [xa : 5; ,f (r,{) L6094381,estimateln 2 with a third-orderNewton's interpolatingpolynomial polynomial, Eq (1510)with n : 4, is Solution The third-order f z @ ) : b r * b z @- x r ) * b z - x r ) ( , r - x ) * b q ( x- , r r ) ( x - x 2 ) ( x - x 3 ) The first divided differences the problem ale [Eq (1515)] for 1386294 O - :04620981 : -_, "f["r:rr] 4

00:00:00.000 00:00:00.000 00:00:00.000

f [4, x2]:

-_ l -

00:00:00.000

:02027326 6-4 1609438 119r'759 :01823216 5-6

The query in listing 6 will return the first and last day of every quarter from January, 2000, until December, 2024.

/[t3, t2, -r1] :

f f x a , x 3 ,x 2 ) :

DECLARE @Date datetime SELECT @Date = '2000-01-01 00:00:00.000' SELECT DATEADD(qq,number,@Date),DATEADD(qq,number+1,@Date)-1 FROM dbo.Numbers WHERE number < 100

: -002041100

.

2000-01-01 2000-04-01 2000-07-01 .... .... 2024-10-01

J ' [ x + , r t - t 2 ,x 1 ] :

: 0007865529

00:00:00.000 00:00:00.000 00:00:00.000 2000-03-31 00:00:00.000 2000-06-30 00:00:00.000 2000-09-30 00:00:00.000

f (xi)

00:00:00.000

4 6 5

0 386294 791759 609438

2024-12-31 00:00:00.000

T h e r e s u l t sf o r / ( x r ) , f [ x z , x t ] , f [ x t , x z , x 1 ] , a n d f l x q , x z , x 2 , t l l r e p r e s e ntth e b1, coefficients b2,b3, andb',, respectively, Eq (l-510) of Thus,the interpolating cubicis -f:(r) :0 + 0 4 6 2 0 9 8 1 ( x l ) - 0 0 5 1 8 7 3 1 l (-" t l ) ( x - 4 ) + 0007865529(x l)(x - 4)(r - 6)

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