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left(
t
right)
=
a_0
+
a_1 t
+
a_2 t^2
+
dotsaxis 
+
a_j t^j
+
dotsaxis
+
a_p t^p</annotation>
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   <text:p text:style-name="equation"/>
   <text:p text:style-name="equation">Therefore, sum of errors squared is</text:p>
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           <mn>2</mn>
          </msup>
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        </mrow>
        <annotation encoding="StarMath 5.0">s
=
sum from i=1 to n
left(
f
left(
t_i
right)
-
f_i
right)^2</annotation>
       </semantics>
      </math>
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   <text:p text:style-name="equation">Minimize wrt<draw:frame draw:style-name="fr1" draw:name="Object4" text:anchor-type="as-char" svg:width="1.3075in" svg:height="0.2102in" draw:z-index="3"><draw:object>
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         <mi>,</mi>
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         </msub>
         <mi>,</mi>
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         <mi>,</mi>
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          <mi>j</mi>
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         <mi>,</mi>
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          <mi>a</mi>
          <mi>p</mi>
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over
{partial a_j}
=
sum from i=1 to n
2
left(
f
left(
t_i
right)
-
f_i
right)
{partial f
left(
t_i
right)
}
over
{partial a_j}
=
sum from i=1 to n
2
left(
a_0
+
a_1 t_i
+
dotsaxis 
+
a_j t^j_i
+
dotsaxis
+
a_p t^p_i
-
f_i
right)
t^j_i


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matrix
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t^j_i #
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t^{0+1}_i #
t^{1+1}_i #
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t^{j+1}_i #
dotsaxis #
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dotsvert  #
dotsdown  #
dotsvert    #
dotsdown  #
dotsvert  ##
t^{0+j}_i #
t^{1+j}_i #
dotsaxis #
t^{j+j}_i #
dotsaxis #
t^{p+j}_i ##
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dotsvert  #
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dotsvert    #
dotsdown  #
dotsvert  ##
t^{0+p}_i #
t^{1+p}_i #
dotsaxis #
t^{j+p}_i #
dotsaxis #
t^{p+p}_i
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right]
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matrix
{
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a_1 ##
dotsvert  ##
a_j ##
dotsvert  ##
a_p 
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right rbrace
=
sum from i=1 to n
left lbrace
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        <annotation encoding="StarMath 5.0">sum from i=1 to n
left[
matrix
{
t^0_i #
t^1_i #
t^2_i ##
t^{0+1}_i #
t^{1+1}_i #
t^{2+1}_i ##
t^{0+2}_i #
t^{1+2}_i #
t^{2+2}_i
}
right]
left lbrace
matrix
{
a_0 ##
a_1 ##
a_2
}
right rbrace
=
sum from i=1 to n
left lbrace
matrix
{
f_i t^0_i ##
f_i t^1_i ##
f_i t^2_i
}
right rbrace





</annotation>
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   <text:p text:style-name="equation">The matrix is ill conditioned when t’s are large. Help by shifting the origin to the middle of the domain of t’s.</text:p>
   <text:h text:style-name="Heading_20_1" text:outline-level="1">Roots of equation</text:h>
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        <annotation encoding="StarMath 5.0">f
left(
t
right)
=
a_0
+
a_1 t
+
a_2 t^2
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=
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over
{2a}
=
{-a_1 +- sqrt {a^2_1 - 4a_2 a_0} }
over
{2a_2}
</annotation>
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