What our customers say...

Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences:

I appreciate that it is basic…It has helped me tremendously…
Derek Brennan, IL

This algebra software has an exceptional ability to accommodate individual users. While offering help with algebra homework, it also forces the student to learn basic math. The algebra tutor part of the software provides easy to understand explanations for every step of algebra problem solution.
Miguel San Miguel-Gonzalez, Laredo Int. University

I have never understood algebra, which caused me to struggle, and ended up making me hate math. Now that I have Algebrator, math no longer seems like a foreign language to me. I enjoy attending my Math class now!
Kevin Porter, TX

Algebrator is a great product. I love how easy it is to use, and how easy it makes algebra seem.
Candice Murrey, OR

My study methods have never been quite good; that is way I always look for an easier learning method, and I strongly recommend the Algebrator.
Patrick Ocean, FL




FREE SIMULTANEOUS QUADRATIC EQUATION SOLVER
free worksheet solving addition and subtraction equations , slope formula, Quadratic formula, slope intercept formula sheet , sqare roots of integer number , solving 3rd order quadratic equations , graph equation basic algebra help , algebra easy finding greatest common denominator , non linear equation solver unknowns equations , rational expressions on-line calculator , algebra help calculate greatest common denominator , basic algebra power fraction , simplifying exponential expressions variable in the exponent , first order linear partial differential equation solutions ebook , How to solve Non linear Equation solving

Thank you for visiting our site! You landed on this page because you entered a search term similar to this: free simultaneous quadratic equation solver. We have an extensive database of resources on free simultaneous quadratic equation solver. Below is one of them. If you need further help, please take a look at our software "Algebrator", a software program that can solve any algebra problem you enter!


  (28)
Now the weighted sum-squared error will be  
 \begin{displaymath}
E \eq \sum_i w_i {e_i}^2\end{displaymath} (29)
Following the method of the last section, it is easy to show that the x which minimizes the weighted error E of (29) is the x which satisfies the simultaneous equations  
 \begin{displaymath}
\left\{ \sum_{i} w_i 
\; \left[ 
\begin{array}
{c}
 -c_i \\ ...
 ...{array}
{c}
 v \\  
 0 \\  
 0 \\  
 \vdots \end{array} \right]\end{displaymath} (30)
Choice of a set of weights is often a rather subjective matter. However, if data are of uneven quality, it cannot be avoided. Omitting w is equivalent to choosing it equal to unity.

A case of common interest is where some equations should be solved exactly. Such equations are called constraint equations. Constraint equations often arise out of theoretical considerations so they may, in principle, not have any error. The rest of the equations often involve some measurement. Since the measurement can often be made many times, it is easy to get a lot more equations than unknowns. Since measurement always involves error, we then use the method of least squares to minimize the average error. In order to be certain that the constraint equations are solved exactly, one could use the trick of applying very large weight factors to the constraint equations. A problem is that ``very large'' is not well defined. A weight equal 1010 might not be large enough to guarantee the constraint equation is satisfied with sufficient accuracy. On the other hand, 1010 might lead to disastrous round-off when solving the simultaneous equations in a computer with eight-digit accuracy. The best approach is to analyze the situation theoretically for $w \rightarrow \infty$.

An example of a constraint equation is that the sum of the xi equals M. Another constraint would be x1 = x2. Arranged in a matrix, these two constraint equations are  

 \begin{displaymath}
\left[
\begin{array}
{rcrccc}
 -M & 1 & 1 & 1 & 1 & \cdots \...
 ...t] 
\eq \left[
\begin{array}
{c}
 0 \\  
 0 \end{array} \right]\end{displaymath} (31)
We write a general set of k constraint equations as  
 \begin{displaymath}
{\bf G} 
\left[
\begin{array}
{c}
 1 \\  
 x_1 \\  
 x_2 \\ ...
 ...eq \left[
\begin{array}
{c}
 0 \\  
 \vdots \end{array} \right]\end{displaymath} (32)
Minimizing the error as $w \rightarrow \infty$ of the equations
\begin{eqnarraystar}
\sqrt{w} {\bf Gx} &\approx & {\bf 0} \nonumber \\  {\bf Bx} &\approx & {\bf 0} \end{eqnarraystar}
is algebraically similar to minimizing the error of Bx $\approx$ 0. The rows of $\sqrt{w} {\bf G}$ are just like some extra rows for B. The resulting equation for x is  
 \begin{displaymath}
\left\{ \sum^n_{i = 1} 
\left[ 
\begin{array}
{c}
 -c_i \\  ...
 ... 
\begin{array}
{c}
 v \\  
 0 \\  
 \vdots \end{array} \right]\end{displaymath} (33)
Now we will take all the wi to equal $1/\varepsilon$ and we will let $\varepsilon$tend to zero. Also let
      \begin{eqnarray}
{\bf x} &= & {\bf x}^{(0)} 
+ \varepsilon {\bf x}^{(1)} 
+ \var...
 ... \varepsilon {\bf v}^{(1)} 
+ \varepsilon^2 {\bf v}^{(2)}
+ \cdots\end{eqnarray} (34)
(35)
With this, (33) may be written  
 \begin{displaymath}
\left( {\bf B}^T {\bf B} + {1 \over {\varepsilon}} {\bf G}^T...
 ...\cdots) 
\eq {\bf v}^{(0)} + {\bf v}^{(1)} \varepsilon + \cdots\end{displaymath} (36)
Identify coefficients of powers of $\varepsilon$
      \begin{eqnarray}
\varepsilon^{-1} & : & {\bf G}^T {\bf Gx}^{(0)} = {\bf 0}
\\  \...
 ...  \varepsilon^1, \varepsilon^2 & : & \hbox{not required} \nonumber\end{eqnarray} (37)
(38)
Equation (37) is m equations in m unknowns. It will automatically be satisfied if the k equations in (32) are satisfied. Equation (38) appears to involve the m unknowns in ${\bf x}^{(0)}$ plus m more unknowns in ${\bf x}^{(1)}$. In fact, we do not need ${\bf x}^{(1)}$;the k unknowns  
 \begin{displaymath}
\lambda \eq {\bf Gx}^{(1)}\end{displaymath} (39)
will suffice.

Arranging (38) and (32) together and dropping superscripts, we get a square matrix in m + k unknowns.  

 \begin{displaymath}
\left[ 
 \begin{array}
{c}
 {\bf B}^T {\bf B} \\  
 \vbox{\h...
 ...t[ 
\begin{array}
{c}
 v \\  0 \\  
 \vdots \end{array} \right]\end{displaymath} (40)

Equation (40) is now a simultaneous set for the unknowns ${\bf x}$and $\lambda$. It might also be thought of as the solution to the problem of minimizing the quadratic form

\begin{eqnarraystar}
E &= & [{\bf x}^T \quad \lambda^T]\left[
\begin{array}
{cc}...
 ... \lambda^T {\bf Gx} + {\bf x}^T {\bf G}^T 
 \lambda \nonumber \end{eqnarraystar}
and since we can always transpose a scalar,  
 \begin{displaymath}
E \eq {\bf x}^T {\bf B}^T {\bf Bx} + 2\lambda^T {\bf Gx}\end{displaymath} (41)

 

Order on CD or Instant Download
Only   $74.99   $39.99
This offer is good until
midnight March 22nd only!
How can Algebrator Help YOU?

  • It solves any problem from your textbook
  • It gives you all the steps, not just solutions - just like a teacher!
  • Algebrator is your personal 24/7 math tutor that costs less than one hour of live tutoring
  • When you don't understand a step, it gives you an explanation
  • You get your homework done in minutes, and you learn algebra
Algebrator can solve problems in all the following areas:
  • simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
  • factoring and expanding expressions
  • finding LCM and GCF
  • operations with complex numbers (simplifying, rationalizing complex denominators...)
  • solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
  • solving a system of two and three linear equations (including Cramer's rule)
  • graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
  • graphing general functions
  • operations with functions (composition, inverse, range, domain...)
  • simplifying logarithms
  • basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
  • arithmetic and other pre-algebra topics (ratios, proportions, measurements...)
  • NEW! Linear Algebra (operations with matrices, inverse matrix, determinants...)





Instant Bonus: Receive thousands of problems pre-entered
by Algebrator's users if you buy the software now!
Order on CD or Instant Download
Only   $74.99   $39.99
This offer is good until
midnight March 22nd only!