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Newton’s Interpolation is a powerful mathematical technique used to find a polynomial function that passes exactly through a given set of data points. It is highly valued in computational mathematics because it allows you to add new data points easily without recalculating the entire polynomial from scratch. 1. Understand the Core Concepts

Newton’s interpolation relies on divided differences to construct a polynomial step-by-step. The Interpolating Polynomial For a given set of data points , the Newton’s divided-difference interpolating polynomial is expressed as:

Pn(x)=a0+a1(x−x0)+a2(x−x0)(x−x1)+…+an(x−x0)(x−x1)…(x−xn−1)cap P sub n open paren x close paren equals a sub 0 plus a sub 1 open paren x minus x sub 0 close paren plus a sub 2 open paren x minus x sub 0 close paren open paren x minus x sub 1 close paren plus … plus a sub n open paren x minus x sub 0 close paren open paren x minus x sub 1 close paren … open paren x minus x sub n minus 1 end-sub close paren Where the coefficients correspond to the calculated divided differences: Divided Difference Formulas

The divided differences are calculated recursively using the following definitions: Zeroth divided difference: First divided difference: Second divided difference: 2. Construct the Divided Difference Table

To make calculations seamless, arrange your data into a systematic table. Let’s walk through an example using 3 data points ( , which will yield a quadratic polynomial): Step-by-Step Table Calculation Calculate the First Divided Differences:

f[x0,x1]=7−32−1=4f of open bracket x sub 0 comma x sub 1 close bracket equals the fraction with numerator 7 minus 3 and denominator 2 minus 1 end-fraction equals 4

f[x1,x2]=21−74−2=142=7f of open bracket x sub 1 comma x sub 2 close bracket equals the fraction with numerator 21 minus 7 and denominator 4 minus 2 end-fraction equals fourteen-halves equals 7 Calculate the Second Divided Difference:

f[x0,x1,x2]=f[x1,x2]−f[x0,x1]x2−x0=7−44−1=33=1f of open bracket x sub 0 comma x sub 1 comma x sub 2 close bracket equals the fraction with numerator f of open bracket x sub 1 comma x sub 2 close bracket minus f of open bracket x sub 0 comma x sub 1 close bracket and denominator x sub 2 minus x sub 0 end-fraction equals the fraction with numerator 7 minus 4 and denominator 4 minus 1 end-fraction equals three-thirds equals 1 The Resulting Table The coefficients we need (

) are always found along the top diagonal of the calculated columns: 0th0 raised to the th power 1st1 raised to the st power Divided Diff 2nd2 raised to the nd power Divided Diff 1 3 ( 4 ( 2 7 1 ( 7 4 21 3. Build the Final Polynomial Now substitute your top diagonal coefficients ( ) into the Newton formula:

P2(x)=a0+a1(x−x0)+a2(x−x0)(x−x1)cap P sub 2 open paren x close paren equals a sub 0 plus a sub 1 open paren x minus x sub 0 close paren plus a sub 2 open paren x minus x sub 0 close paren open paren x minus x sub 1 close paren

P2(x)=3+4(x−1)+1(x−1)(x−2)cap P sub 2 open paren x close paren equals 3 plus 4 open paren x minus 1 close paren plus 1 open paren x minus 1 close paren open paren x minus 2 close paren Simplify the Expression

P2(x)=3+4x−4+(x2−3x+2)cap P sub 2 open paren x close paren equals 3 plus 4 x minus 4 plus open paren x squared minus 3 x plus 2 close paren

P2(x)=x2+x+1cap P sub 2 open paren x close paren equals x squared plus x plus 1 4. Evaluate an Unknown Value You can use the constructed polynomial to estimate for any value of within or near your data range. Example Calculation Let’s approximate the value of

P2(3)=(3)2+3+1=9+3+1=13cap P sub 2 open paren 3 close paren equals open paren 3 close paren squared plus 3 plus 1 equals 9 plus 3 plus 1 equals 13 5. Visualize the Interpolation

The chart below shows how the resulting quadratic polynomial

smoothly tracks through all three given data coordinates and provides the interpolated value at ✅ Summary of Method The final interpolating polynomial for the data points is , yielding an estimated value of at

If you want to apply this to a specific dataset, please share: Your data points or coordinates The unknown target value you need to approximate

intervals are equally spaced (which allows using Newton’s Forward/Backward methods instead)