Using solver function

Pressure. We will use Solver function to find the value of the cell Temperature that provides the value of the cell Pressure as atmospheric pressure that is 760 mmHg. 

In step 3, a multiline-fitting program was run to optimize the pK a values to minimize the sum of residual squares between calculated and observed mobilities from Eq. (17). Figure 2 shows an example of the MS Excel spreadsheet for pK a calculation. The solver function of MS Excel could be used to perform the multiline-fitting analysis. 

The spreadsheet could be programmed to use the TOOLS-SOLVER function to automate the selection of the pressure-gradient parameter, cell B8, to drive the mean nondi-mensional velocity to 1.0. However, our experience in using this spreadsheet is that the iteration proceeds more efficiently if one simply guesses the value of the parameter and watches the value for the calculated mean velocity in cell E9. Making a series of successively more accurate guesses in cell B8 could be used to solve the problem for a new aspect ratio in just a few seconds. Furthermore it is fun to watch the iterations spin by ... 

Rearrange Equation A to solve for lAg+] as a function of [H+] or use solver to find [Ag4 ] as a function of [H+], We will use the algebraic solution, which is easy for this exercise. Multiply both sides by [Ag+] and solve ... 

The lines that follow the initial % MATLAB comment lines in fixedbedreact.m set up default values for the seven optional parameters. Then we prepare for the MATLAB IVP solver ode.. . that solves our problem by using the function dydt to evaluate the right-hand side of our IVP (4.22). Having solved (4.22) we plot two curves of the solution to the two joint DEs. 

Use Solver., to minimize the sum of squares of residuals (the target cell) by changing the coefficients of the function (the changing cells). 

Alternatively, you can use Solver to fit data to any analytical function of your fancy, then use the fitting parameters to compute the function at any given point within the range covered. 

Kinetic and electrochemical data, respectively, were fitted to eqs 28 and 29. Non-linear least-square fits of the observed rate constant and the formal redox potential versus [Ml were carried out using the Solver Function in Microsoft Excel-98. Sums of deviation-squared values were minimized by varying k, kivir, Ko, Kred, Eo, a and b in eqs 28-30. The ratios Yox/y ox and Yred/y rcd were assigned a value of 1. Additional limitations and constraints imposed on the adjustable parameters to improve fitting are discussed above. 

Figure 4.11 Calculation of the 2s orbital energy in hydrogen with the sto-3g ls> basis set, Table 1.6, for the 2s Slater function rendered orthogonal to the sto-3g ls> function. The initial calculation returns a poor estimate of the energy terms and 2 for the minimization condition on the least-squares integral of Chapter 3. Optimization based on the minimization of the energy, using SOLVER on the Slater exponent, returns closer agreement with the exact results.

Using the function Solver (Data) in MS Excel, we obtain C id,emax 3.5-10" mole-1" or 35.0 mg of calcite per liter. 

To obtain a standard deviation to express an experimental error with a confidence area, procedures to obtain standard deviations of parameters of general functions are indispensable. These are, however, quite complex. With the aid of spreadsheet software, the standard deviations can be obtained relatively easily. In this case, a convenient way to obtain the standard deviation is to apply the Solvstat macro to the spreadsheet. Doing so is easy and provides reliable results. First, a spreadsheet to obtain regression coefficients by using Solver must be prepared. Then the Solvstat add-in can be used to elucidate standard deviations of regression coefficients. ... 

To calculate weakest preconditions, we define the function WV S). >p, and consider the postcondition True. The definition of yV P S).ip is mostfy standard, except that we consider that expressions can raise exceptions and prevent proper termination. We, therefore, use auxiiiary functions that determine when a expression or a command terminates. Definitions are provided in [13], afong with calcu-fations for the quadratic solver. 

Figure 7.7 callmultiode.m (script M-file) that defines integration-related parameters and calls the MATLAB ode45 solver using a function handle (i.e., multiode) to multiode function. 

Use the MATLAB ode45 solver to solve Problem 7.3. First, define the derivative using the function M-file named Pr7 4.m. Second, write the calling code (main code) that defines the initial value of y, the time interval, calls the MATLAB ode45 solver, defines the exact... 

The models verified by model checkers are usually very detailed and explicit -equivalent to executable programs. However, and this is not widely understood, infinite bounded model checkers can be applied to rather abstract descriptions that use uninterpreted functions to hide detail. This is feasible because the underlying SMT solvers provide effective automation for this theory. Properties can be attached to the uninterpreted functions by means of axioms supplied directly to the SMT solver or, indirectly, by synchronous observers attached to the model supplied to the model checker (Rushby 2009a). 

Nonlinear least-squares analysis can be used to obtain best fit values of the unknown parameters in a nonlinear rate model. Elementary nonlinear regressions can be performed using the SOLVER function in EXCEL. 

There are two ways in which to use Solver for nonlinear equations. The direct way is to set up the nonlinear equations eis constraints with no objective function. The other way is to set up the spreadsheet to compute the sum of squares of residuals and use Solver to minimize this (without any constraints). The latter method is used in the following spreadsheet, where the feed consists only of component A with Qq = 1. The volumetric flow rate is 50 gmol/s, and the reactor volume is 100 L/s. The equations are rearranged in the form f(x) = 0 so that the left-hand sides are residuals whose value at a solution is zero (within tolerance). The initial guess for all concentrations is 0.5 gmol/L. 

First, solve this problem analytically by solving the constraint for xf and substituting this into the objective function. Then differentiate the objective function (the only remaining variable is x, set the derivative to zero, and find X2. Use the value for X2 to find the value(s) for x,. Next use Solver to find the solution(s). Use a starting point of [1, 1] and then [-1, -1] and see what solutions Solver finds from these starting points. 

As discussed in Section 1.6.1 the microkinetic model may be solved as a system of ODEs or non-linear algebraic equations using the steady-state assumption. It turns out that, regardless of which approach you want to use, the function that must be passed to an ODE solver or numerical root-finding method is the same Here, the more general case of the ODE system is chosen. Note that we named the previously defined function get ratesQ. 

The h-2h calculation needed for the Richardson type extrapolation is also the calculation needed to estimate the error in the solution of a boundary value problem as discussed in the previous section. Thus it seem appropriate to combine these into an eigenvalue solver and such a function has been coded as the function odebveveO which is also available with the require obebv statement. An example of using this function for a higher order eigenvalue and function of the same constant potential problem is shown in Listing 11.10. The difference from Listing... 

A search is then made by varying x and xf simultaneously (e.g. using a spreadsheet solver) to solve the objective function (see Section 3.9) ... 

However, the most appropriate value of the rate constant for each model needs to be determined. This can be determined, for example, in a spreadsheet by setting up a function for R2 in the spreadsheet and then using the spreadsheet solver to minimize R2 by manipulating the value of kA. The results are summarized in Table 5.6. 

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