Multivariate polynomial fit: Decay heat removal system and pectin degrading Fe3O4‐SiO2 nanobiocatalyst activity

Abstract Herein, multivariate Lagrange's interpolation polynomial (MLIP) and multivariate least square (MLS) methods are used to derive linear and higher‐order polynomials for two varied applications. (1) For an effective fabrication of Pectin degrading Fe3O4‐SiO2 Nanobiocatalyst activity (IU/mg). Here, the three parameters namely: pH value, pectinase loading and temperature as independent variables are optimized for the maximal of anobiocatalyst activity as a dependent variable. (2) For a passive system reliability estimation of decay heat removal (DHR) of a nuclear power plant. The success criteria of the system depend on three types temperature that do not exceed their respective design safety limits and are considered as dependent variables and 14 significant parameters were used as independent variables. Statistically, the validation of these multivariate polynomials are done by testing of hypothesis. Comparative study of the proposed approach gives significance results in the first application have the optimum conditions for maximum activity using linear MLIP method is: 58.64 with pH = 4, pL = 250 and Temp = 4°C. The maximum activity using second order MLIP method is 59.825 and method of MLS is 59.8249 with the optimized values of an independent variables pH = 4, pL = 300 and Temp = 8°C depicted in Table 1. In DHR system, the significance results are obtained and depicted in Table 2.


| INTRODUCTION
Interpolation is one of the most important technique in numerical analysis for the solutions of many real-time applications in science and engineering [1]. It is frequently used to approximate the polynomials for a given set of data. In most of the cases, an unique analytical expression can be derived by interpolation of simple algebraic expressions such as: polynomials, exponential, logarithmic functions and power curves which enable us to represent a relationship between dependent over independent variables [2]. However, for huge data, derivation of interpolating polynomial is very tedious and more time consuming. Alternatively, the regression analysis is used to fit a relationship between the simulated values to the actual data set. Clearly, it expresses a polynomial representation depending on the existence of a relationship among various independent variables. In some cases, the observed data are imprecise while using the uncertainty theory to design the uncertain regression model of these observed data as uncertain variables [3]. Also, estimating the parameters of uncertain regression model using principal of least square. Usually, the least square method is used in linear regression for a best fit of a line or curve. This method calculates to minimize the sum of the squares of offsets for pairs of data points from a curve. Squares of these offsets are called residuals [4,5].
For any simulation study in science and engineering, it is not always feasible to develop an exact representation of input and output response. Also, if the problem is of higher dimension and involves large amount of data, validation becomes difficult. Hence, an universally acceptable method for a real-time application is never possible [2]. In such a case, the linear regression is used for fitting as a special kind of multiple linear regression models. This is a popular statistical technique that can be widely used to derive the linear relationship between variables. Most -173 frequently, the practical data have non-linear or curvilinear function representation the polynomial regressions methods are appropriate. Also, the parameters of the polynomial regression are determined by method of least square. This method having a problem of computing the correlation coefficient among variables of the polynomial regression that possess an ill-conditioned/multi-collinearity. It shows the computations of matrix operations in least square method approaches to singular value. Thus, the multi-collinearity possess an unreliable of the parameter estimation in the polynomial regression over the large variances [6,7]. There are many steps focused to fit better regression models to understand the relationship between variables of the data set, deriving relationship using prediction models, error estimation and inferences about results of the prediction model to fit actual data set and this will helps to determine the correlation between the variables. Various regression analysis are used for data fitting like: linear and multivariate linear regressions, polynomial and multivariate polynomial regressions, and quadratic polynomial regressions etc. [8]. The other methods like, Response Surface Methodology (RSM), central composite design experiment are used to fit a mathematical expression between dependent over independent variables [9]. Generally, the multivariate polynomial regression is extended from the simple regression model [10]. The disadvantages of the multivariate regression models does not have scope for the smaller data set with high-levels of mathematical computations. Sometimes, the output of the model cannot be easily interpreted and the error outputs are not identical. Reasonably for this situation, most of the researchers preferred method of least square and method of least absolute value to fit best curve fit. Also, to avoid ill-conditioned situation the multivariate Lagrange's interpolating polynomial (MLIP) method is preferred as in view of its simplistic approach [11], with error approaches to zero.
The objective of the present study is to fit a linear and higherorder polynomials by MLIP method and method of multivariate least square (MLS) [10] for real-time applications. Statistically, the validation of derived polynomials by Chi-square testing for goodness of fit is applied [12], and the validation of huge number of polynomials follows large sample test.
Herein, Section 2 discusses the related works. Schematic representation with advantages of the proposed approach are explained in Section 3. Section 4 discusses the MLIP method and method of MLS is presented in Section 5. Proposed approach is applied for two real-time applications, the statistical validations with comparative results are discussed in Section 6. Short summary of Matlab computations with statistical validations are presented in Section 7 and Section 8 respectively. Finally, the results and conclusions are summarized in Section 9.

| RELATED WORKS
Many of the real-world applications have multiple input parameters and output responses. In such applications, multivariate polynomial is derived for m input variables with degree p over p + m combinations of p distinct data points [13]. These methods are mathematically discussed using set of input arguments in [14], the derived polynomials by rational numbers [15] and Newton form of multivariate polynomial interpolation are discussed in [2]. Regression model is more useful for predicting the forecasting technique of future values, data mining and many applications in engineering domains. In addition, the Bayesian latent factor regression for multivariate functional data with variable selection in [16]. In chemical engineering application, the optimization of acid dye bio-sorption of brewery waste biomass using RSM is considered. The percentage of removal efficiency is significantly influenced by time, pH, adsorbent dosage with initial dye concentration and multivariate polynomial methods are adopted subsequently [17]. One-Factor At-a-Time (OFAT) procedure is followed by RSM with canonical analysis was used to optimize the immobilization parameters such as: glutaraldehyde concentration, pH value, temperature and pectinase loading for the effective fabrication characterization and application of pectin degrading Fe 3 O 4 -SiO 2 nanobiocatalyst were discussed in [9,18]. The other similar real-world application where multivariate polynomial is used as characterization of the relationship between strains and drilling depth in a metallurgical plant is discussed in [19]. Similarly, for the estimation of reliability of a passive decay heat removal system of a nuclear power plant (NPP), one simple approach to avoid rigorous computational exercise is by adopting RSM [20]. Also, the regression analysis and curve fitting models are used to approximate the analytic expression of given discrete data. However, the linear and non-linear regression models are widely used for passive system reliability estimation [21]. Multivariate polynomial fit based on the RSM produces reasonably realistic estimation. The results are not only computationally efficient but also consistent. The accuracy and efficiency of this approach is compared to the direct Monte-Carlo simulation [22]. The success criteria of the system is taken from [23] and the failure probability evaluation of the passive system analysis is verified with other techniques such as fuzzy Monte-Carlo simulation [24]. Passive systems involves in new generation nuclear design for safety-critical functions mainly concentrate to improve safety and reduce human errors with low cost as well as avoid dependence on the external power source. Moreover, the reliability estimation of passive system is theoretically more reliable as compared to active components. But, uncertainty involved in this system to be very large. Therefore, we necessarily need to quantify the uncertainty. Some uncertainties may happen due to insufficient information, such a case the fuzzy set theory is more appropriate instead of probability concepts [21]. For instance, to handle large amount of data with finite number of thermal-hydraulic code runs, the regression analysis was performed. To perform an effective regression models, it is necessary to find the effect of every input parameter that goes to into the system [25].

| Scope of the proposed study
From the above detailed review, the proposed study addresses the research gap for polynomial contributions in real-time applications.
1. To develop a relationship between several independent and dependent variables, 2. Multivariate mathematical methods are preferred, 3. To validate the multivariate polynomials, 4. Testing of hypothesis is applied, 5. Different real-time applications are discussed, 6. Comparative study of the proposed approach.

| SCHEMATIC REPRESENTATION OF THE PROPOSED APPROACH
This section discusses the detailed workflow, proposed approach, merits and demerits of the mathematical models for real-time applications.

| Context and solution technique
In this section, the aim of the study, mathematical methods, procedure for deriving polynomials and validations are discussed in detail.
Aims: The advantages of multivariate polynomial methods to develop the relationship between dependent and several independent variables which will be used to identify the significant contributor of the output in an application. Methods: For the above context, the mathematical methods are very much essential for solving real-time applications. Therefore, MLIP and method of MLS are preferred to fit a linear and higher-order polynomials for the above real-time applications. The necessary and sufficient condition for the uniqueness of MLIP method is that the sample square matrix is non-singular [13,26]. In practical application, if the sample matrix is singular then the MLIP method fails (ill-conditioned).
In such a case, the method of MLS is preferred to quantify the polynomial coefficients [4]. Here, Microsoft Mathematics tool and Matlab programming language were used for performing high-level computations of this approach. Thus, the list of factors are ordered and is used to fit a multivariate polynomial shown in Figure 1. Procedure: A detailed procedure for deriving multivariate polynomials with their statistical validation of the applications are shown in Figure 2. The results were carried out and we concentrated more on stress in the statistical validation to assess the practical significance of the problem. Polynomial Validation: To verify and validate the multivariate linear and/or higher-order polynomials by testing of hypothesis is preferred. Moreover, the contribution of the statistical analysis is to assess the practical significance of the testing results. Thus, the derivation of multivariate polynomials depends on each sample square matrix of the above methods. More clearly, the sample square matrix is selected based on the size of sample runs. The above mathematical methods are directly applied for one and only one sample matrix to derive polynomial which is possible only when the number of independent variables involved in the practical application is same as the number of sample runs (each non-singular matrix of the method derives one multivariate polynomial). Otherwise, if the size of sample runs is more as compared to number of independent variables, then more than one polynomial computation is applicable for the simulation study. Therefore, the above methods are applicable for polynomial computations while the selection of sample square matrices depending on the combination among all sample runs, that is the size of the sample runs N is more as compared to number of variables K (selection of K multivariate vectors out of sample size N) then counting the selection of sample square matrices is Finally, validation of the derived polynomial follows small sample test, if the number of sample square matrices are less or equal to 30. Otherwise, the method follows large sample test. In small sample test, the χ 2 testing for goodness of fit is performed in which the experimental outputs are taken as observed frequency (O i ). The simulated outputs are the expected frequencies (E i ) calculated from derived polynomials by both mathematical methods which are treated as output arguments depending on the corresponding input data set. To validate these polynomials, the following hypothetical assumptions are: 1. Testing the hypothesis against the derived polynomials of the application which depends on the number of matrices (size say: n). Also, the null hypothesis of the problem is decided based on the sample square matrices which are less than or equal to 30. 2. Under H 0 : 'The multivariate polynomial fit is good one'.
That is, there is no significant difference between the observed and expected frequencies. 3. Otherwise H 1 : 'The multivariate polynomial fit is not good one'. That is, there is a significant difference between the observed and expected frequencies. 4. Test statistic: Under H 0 , the test statistic is given by The mathematical approach for the solutions of above case study are shown in Figure 3.

| Merits and De-merits: multivariate polynomial methods
The next section outlines the merits and demerits of the mathematical methods in practical situations.

| Merits
The merits of the proposed work are discussed in the following.

Mathematical relationship between variables response is
developed as a multivariate linear and higher-order polynomials using MLIP and/or MLS methods. 2. Extrapolating beyond the range of input arguments, the input and output relationship is developed without repeating the experiments (simulation). 3. The polynomial representation is quickly solvable and consumes less time to obtain the desired results. 4. The verification and validation of the polynomials can be carried out using χ 2 testing for goodness of fit. Otherwise, it follows large sample test. 5. The multivariate polynomial methods are more helpful for further reliability quantifications in safety-critical applications.

| Demerits
The demerits of the method is given below.
1. When the application consists of huge number of parameters, it is very tedious to develop an exact relationship between input and output arguments. The unique multivariate polynomial fit derived by MLIP method is discussed in the next section.

| MULTIVARIATE LAGRANGE's INTERPOLATING POLYNOMIAL
Let f(X 1 , X 2 , …, X m ) be multinomial function of m independent variables x 1 , x 2 , …, x m over degree p in [27]. There are (p + m)C p = ρ as number of terms in f, the necessary condition that ρ number of distinct data points f(x 1,i , x 2,i , …, x m, i ) = f i in R mþ1 and f is uniquely defined [8,10]. Then, Where α e i are the coefficients in f. The MLIP function is of the form Here, l i (X ) is a multinomial function of several independent variables X 1 , X 2 , …, X m having the property that X is equal to i th data point, that is X = x i (x 1,i , x 2,i , …, x m,i ). Moreover, l i (X ) satisfies the following condition: -177 The function f i gives system of linear equation The sample square matrix M ¼ x e j i À � with determinant det (M) is non-singular. Therefore, At X = x i in M j (X) has i th row appears twice, then detððM j ðXÞÞ i Þ ¼ 0 for ði ≠ jÞ. Hence, l i ðXÞ ¼ Δ i ðXÞ Δ . The unique multivariate linear and higher-order polynomial is If MLIP method fails, then the method of MLS is preferred.

| METHOD OF MULTIVARIATE LEAST SQUARE
The method of MLS for second-order polynomial over three independent variables has the form Consider m 0 by applying method of MLS to minimize error E mls from the actual k data points.
Take partial derivatives of E mls in Equation (8) with respect to a 0 , b 0 , c 0 , d 0 , e 0 , f 0 , g 0 , h 0 , k 0 , m 0 and equating to zero have the following equations: The set of equations (9) to (18) can be rearranged into matrix form by MUTHUSAMY ET AL. -179 The above Equation (19), can be written as the Vandermonde square matrix over three independent variables. Then, 1 x 9 y 9 z 9 x 9 y 9 y 9 z 9 z 9 x 9 x 2 9 y 2 9 z 2 9 1 x 10 y 10 z 10 x 10 y 10 y 10 z 10 z 10 x 10 x 2 10 y 2 Now, premultiply by Van T on both sides of Equation (21). Then, ðV an T * V anÞ * A mls ¼ ðV anÞ T * Y Therefore, the co-efficient of MLS is computed as The matrix A mls gives the coefficient of the secondorder multivariate polynomial. This way, MLS can be used to fit a polynomial for real-time applications with large number of parameters. In the next section, application of both mathematical methods are demonstrated through case studies.

| CASE STUDY
In this section, two different case studies were discussed. The approach for the relationship between case studies and mathematical methods as graphically shown in Figure 3. The detailed workflow for fitting linear and higher-order multivariate polynomials were shown in Figure 2. Moreover, the solution methodology described by three steps to solve the real time applications are listed in the following order.   The activity of pectin degrading Fe 3 O 4 -SiO 2 nanobiocatalyst was determined by measuring the reducing sugar produced as a result of reaction between pectinase and pectin. About 500 μL of varying concentrations of pectinase (free pectinase and bound pectinase separately) solution was prepared using 0.1 M acetate buffer and this was added to 1.0 ml of pectin solution (prepared using 0.1 M acetate buffer) containing 2.0 mg of pectin. The reaction mixture was incubated for 1 h at 50°C under shaking condition. The concentration of reducing sugar (galacturonic acid) in the supernatant was estimated by DNS method as described by Miller (1959) using D-(+)-galacturonic acid monohydrate as standard [18,28]. One unit of pectinase activity (IU/mg) is defined as the amount of galacturonic acid produced (mol) per mg of pectinase per min at pH 4.0 and 50°C. The maximum activity of Fe 3 O 4 -SiO 2 nanobiocatalyst activity was achieved using 10% glutaraldehyde [9]. All experiments were carried out in triplicates and the mean values were calculated to achieve accuracy.

| Mathematical analysis
The aim of the current study is to derive a linear and higher order polynomial by MLIP and method of MLS. For this problem, the One-Factor At-a-Time (OFAT) procedure is followed by RSM and canonical analysis is used to optimize the immobilization parameters. Based on the previous study [9], three parameters were found to be governing and hence these three independent variables namely: pH, pectinase loading and temperature were optimized for their maximal Fe 3 O 4 -SiO 2 nanobiocatalyst activity (IU/mg) which is a dependent variable. The levels obtained from OFAT loading for three independent variables were pH (4.0-6.0); pectinase loading (200-300 μg); and temperature (0-8°C). This experimental analysis is more useful for fitting of polynomials by these mathematical methods.

| MLIP method for Linear Polynomial Computation
The MLIP method is applied to derive a linear multivariate polynomial as similar to previous case study. The results of 20 runs for three independent variables such as, pH value (pH), pectinase loading (pL) and temperature (Temp), as reported in [9] are given in column 1 of Table 1. Out of these 20 sample run, four samples selected in random are (4,300,8), (5,250,4), (4,200,8) and (6, Using Equation (6), a unique linear polynomial derived by the MLIP method is MUTHUSAMY ET AL.
Using this polynomial in Equation (25), expected frequency is obtained for the remaining 16 runs mentioned in Table 1. 6.1.5 | MLIP method for second order polynomials A method for computing linear polynomial is extended for deriving second-order polynomials by MLIP method. For three independent variables, this approach involves a maximum of nine variables with a constant: 1, pH, pL, Temp, (pH � pL), (pL � Temp), (pH � Temp), (pH) 2 , (pL) 2 , (Temp) 2 having the sample square matrix order 10. Therefore, consider a set of data for deriving the second-order polynomial, a randomly selected samples of 10 runs as (5,250,4), (5,250,8), (6,200,8), (4,200,4), (6, 300, 0), (6,250,4), (4,250,4), (6,300,8), (5,200,4) and (6,200,0)  matrix obtained using three independent variables for secondorder polynomial is given by Thus, using Equation (22) We observe that, the second-order polynomial fit by both mathematical methods are closer and the maximum activity values are depicted in Table 1. 6.1.7 | Chi-square testing for goodness of fit To validate the above all polynomials, the χ 2 testing for goodness of fit is applied which consists of the following five steps: Step 1 Null hypothesis H 0 : "The multivariate polynomial fit is good one". That is, there is no significant difference between the observed and expected frequencies. Hence, H 0 : The observed frequency as the original experimental values and its corresponding expected frequency is the simulated outputs calculated from the linear MLIP polynomial using Equation (25), and second order polynomials using Equations (26) and (28) respectively.
Step 2 Alternative hypothesis H 1 : 'The multivariate polynomial fit is not good one'. That is, there is a significant difference between the observed and expected frequencies. Hence, Step 3 Under H 0 , the χ 2 test statistic is: Step 4 The table value of χ 2 distribution for 3 degree of freedom to fit linear polynomial by MLIP method at α = 0.05 level of significance is χ 2 α = 7.815. For both the MLIP and MLS methods, the table for fitting of second-order polynomials at 9 degree of freedom with α = 0.05 level of significance is 16.919.
Step 5 The data yields a value for the Chi-squared test statistic (calculated value) which does not exceed theoretical value for both polynomial methods. Therefore, H 0 is accepted.
The observed and expected frequencies for linear and second-order polynomials are shown in Table 1 and the comparisons are graphically shown in Figure 5. It is inferred from Figure 5 that there is no significant difference between the observed and expected frequencies obtained by methods. Hence, it may be concluded that 'the multivariate polynomial is a good fit'.
In this small sample test, polynomials are derived using only one sample square matrix as four out of 20 sample runs using MLIP method and 10 out of 20 sample runs using MLS method with their polynomial validation is accepted.   Table 1. In this table, the repetitions of 5 th , 8 th , 9 th and 19 th sample runs were the same as second run, also the 11 th run was same as first run. Apart these repetition runs, 15 sample runs were used for computation out of 20 runs to minimize the size of the sample runs which gives less number of sample square matrices. Totally, about 1339 polynomials were derived out of 3003 (minimizes from 184,756 count) sample square matrices for the computations, while by avoiding, 1664 matrices were singular (avoid ill-condition, and approximately 10 −7 precision is considered for the determinant value). Moreover, the normal distribution having mean and standard deviation is: 53.3259 and 3.1192. In addition to this, estimates having 95% confidence limit for mean lie between 53.2828 and 53.3690. The normal curve plot and its average probability plot are shown in Figures 10 and 11.
In this study, all probability plots having the simulated outputs is very closer to the experimental values and inference of testing the hypothesis is accepted.

| Comparison of results
The mathematical results are compared to experimental study in [9], as the empirical polynomials fitted by both methods shows significance results. The maximum activity of Fe 3 O 4 -SiO 2 Nanobiocatalyst obtained using linear MLIP method is: 58.64 with pH = 4, pL = 250 and Temp = 4°C. In the secondorder polynomials, for pH = 4, pL = 300 and Temp = 8°C have a maximal activity by the MLIP method which is 59.825 and by the method of MLS is 59.8249 as depicted in Table 1. For polynomial fitting, the simulation conducted for different sample runs is being follows large sample test. Hence, the statistical validation of each empirical polynomial is closely fitted to the experimental data in order to determine the effect of relationship between variables. Thus, the inference of statistical validation shows that the fitting of multivariate polynomial is good one. In experiment, the maximum activity was at 54.39 IU/mg when pH = 4.0; Temp = 4°C and pL = 250 μg while the canonical analysis was performed for the successful optimization for fabrication of nanobiocatalyst. The glutaraldehyde concentration was fixed as 10% and the independent variables are optimized using OFAT method followed by RSM.

| Application 2: polynomial fit for passive DHR system in NPP
The studies of this application is under NPP and the detailed experiments are discussed in the following which further leads mathematical analysis.

| The DHR problem in NPP
The objective of the problem is to evaluate the reliability of the passive DHR system. The success criteria of the system is that the fuel clad and the structural temperatures viz., Hot Since the execution of the software code to evaluate the temperatures is time consuming, to estimate the reliability of the system using a representative equation, MLIP method is adopted. Using this method, to verify the optimal temperature do not exceed design safety limits. About 50 experimental runs are performed for constructing the polynomial. Fourteen significant parameters used for evaluating the reliability of the system are identified as: -187 A Dhx , A Ahx , T Air , Pre, T p , T s , T n , T d the details of which are explained in [20]. The sample runs are taken from the experimental values is more useful for further mathematical computations while the sample square matrix must be nonsingular. The detailed polynomial computations are discussed in the following.

| MLIP method for Linear Polynomial Computation
The MLIP method is applied to this problem similar to the previous application. The hypothesis is to test the relation between observed frequencies of three dependent variables HPT, CSACHST and SSACHST against the simulated outputs as the expected frequency of the output response from MLIP method. Therefore, to derive the temperature frequency of output responses: f(HPT), f(CSACHST), f(SSACHST) for dependent variables are represented in equations from (29)  In Table 2, the observed frequencies obtained from the actual computational runs and expected frequencies derived using above polynomials are given.  consists of all possible second-order terms for 14 independent variables. Moreover, order of the matrix is too large (order 120� 120) with determinant is very low, in this case the inverse matrix computation is too high (ill-condition). Thus, the higher order polynomials derived by both methods are not possible to attain good results. that is, there is no significant difference between the observed and expected frequencies for HPT, CSACHST, SSACHST.

| MLIP & method of MLS for second order polynomials
Notice that, the observed frequency are the original experimental values depicted in Table 2 and its corresponding simulated output as the expected frequency as the simulated outputs are computed from Equations (29) to (31).
Step 2: Alternative hypothesis H 1 : The multivariate polynomial fit is not good one for three dependent variables. That is, there is significant difference between the observed and expected frequencies.
Step 3: Under H 0 , the χ 2 test statistic is: Step 5: The null hypothesis of the problem cannot be rejected. Therefore, the data yields the inference about 'the multivariate polynomial fit for HPT, CSACHST and SSACHST is good one'. The comparison between the observed and expected frequency temperature for the output response: HPT, CSACHST and SSACHST is depicted in Table 2.
There are only one sample square matrix is considered over 15 (randomly picked) out of 47 runs with their simulated output is closer to the original experimental values. Once a polynomial with a statistically good fit is obtained, the polynomial representation can subsequently be used for deriving the reliability estimate of the system by simulating the input parameters. Monte-Carlo simulation is a widely accepted and consistent method for simulation.

| Normal distribution curve fit
In this problem, an increase of polynomial validation follows a large sample test. The hypothesis of this problem is similar to small sample test discussed in Section 6.2.4.

MLIP method for Linear Polynomial Computation:
Procedure to fit a normal curve is discussed here.

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The normal curves for HPT, CSACHST and SSACHST are shown in Figure 12, Figure 13 and Figure 14, respectively. Furthermore, the average of normal probability plots of HPT, CSACHST and SSACHST for different polynomials per each sample run are shown in Figure 15, Figure 16 and Figure 17 respectively. In all probability plots, the three dependent variables satisfythe respective required design safety limits.

MLIP and method of MLS for Second-Order Polynomials:
For normal probability plots, the second-order polynomial computation depends on the sample square matrix of order 120�120. According to Section 6. In all probability plots, the simulated outputs are very closer to the experimental values and validation for testing the hypothesis is accepted for this problem.

| Comparative study
A set of 47 observed values obtained from the actual experimental runs performed in the system are compared with that of the polynomial estimates and it is seen from the chi-square test that there is a negligible error in the estimated values compared to the experimental values. This has served as a comparative study and has provided the required confidence in the polynomial fit. Moreover, for the 14 parameters being studied for the analysis, the number of experimental runs is found to be statistically sufficient based on Wilk's criteria.

| MATLAB COMPUTATIONS
From two case studies, the computations of multivariate polynomials derived by these mathematical approaches are done using MATLAB programming language. The validations of these two real-time applications are statistically analysed which are depicted in Table 3, Table 4, Table 5, Table 6 and Table 7.  Figure 12 actual execution of the code is expensive. The polynomial can also be used to build the response surface simulation. 5. Practically, in DHR application the number of variable is 14 over three dependent variables derives each linear polynomial. Therefore, this MLIP method fails for second degree polynomial count is C(14 + 2, 2) in which the sample square matrix consists of order 120 is singular. 6. According to Equation (22), the method of MLS fails for DHR application, since the matrix is singular. 7. The MLIP method fails, when the sample run are collinear positions.

| CONCLUSIONS AND FUTURE WORKS
Herein, if the number of sample square matrices (only nonsingular matrices) are less than or equal to 30, it follows a small sample test. This is possible, if the number of variables involved in the application is same as the number of sample square matrices. Also, polynomials were derived for one and only one matrix in which χ 2 testing for goodness of fit is accepted. If the number of non-singular matrices are more (n > 30), it follows large sample test. Otherwise, the sample runs are more as compared to the number of variables; hence, we prefer a combination formula for selection of matrices. In this case, the polynomial validation by a large sample test is applied and the size of the sample is decided by n = number of polyomials � size of sample runs.
Even though, if both method fails when the matrix is singular, in practical scenario the determinant value tends to zero then, the inverse computation is practically not possible. To avoid this situation, we fix the precision of the determinant values in which the matrix is to be considered as non-singular. In DHR problem, this precision is considered as above 10 −7 decimal places (avoid for ill-condition). For the time-consuming, consider a random selection of 18 samples runs for the    In the MLS method, the inverse operation determines the co-efficient of second-order polynomials. It means, a low precision affects the inverse operation ðV an • V an T Þ −1 as well as affects this inverse by the product of transpose matrix ðV an • V an T Þ −1 � V an T (Refer Equation (22)). Therefore, this is to decide if the adoptable precision values classify the matrix as either singular or non-singular. Another disadvantage is that if the number of variables is increased, and the method of MLS is difficult. Practically, the MLIP method is preferred when the values of (Van • Van T ) are too high which leads to affect the inverse and determinant operations of the MLS method. Thus, the MLIP method is more suitable and the MATLAB computation provides good precision for the applications.
In the chemical engineering problem, we have 20C 4 combinations and 20C 10 combinations of sample matrices were used to fit the linear and second-order polynomials. The maximum activity of Fe 3 O 4 -SiO 2 nanobiocatalyst activity achieved using the linear MLIP method is: 58.64 with pH = 4, pL = 250 and Temp = 4. In addition, pH = 4, pL = 300 and Temp = 8 has maximal activity for the fitting of second-order polynomials by the MLIP method is 59.825, whereas in the method of MLS it is 59.82499999 as depicted in Table 1. In the second problem, success criteria of the DHR system are that the fuel clad and the structural temperatures namely: HPT, CSACHST, SSACHST do not exceed their respective design safety limits as verified using multivariate linear polynomials representation over 14 design variables (independent) by MLIP method mentioned in equation from (29) to (31). Furthermore, the higher order polynomials are not suitable, since the number of variables in second order is large, while their values are closer to zero. Also, the derived polynomials are validated by χ 2 testing for goodness of fit not rejected at 5% significance level.
Summary of the study was calculated from the mean and standard deviation and estimates at 95% confidence interval bounds. Thus, the normal probability curve is plotted and the average is determined samplewise for different polynomials. In the first application, the normal probability curve and its average probability plots are shown in Figures 6-11 respectively. Also, for second problem, the normal probability curve and its average probability plots are shown in Figures 12-17 respectively.
In both methods, the number of independent variable is small then the second-order polynomial computation by either using the MLIP or the method of MLS gives similar polynomials in Equations (26) and (28) as discussed in the first application. The variables take integer values from both methods which control the inverse operation. In the MLS method, the sample square matrix is obtained from partial derivatives (Vandermonde matrix of three variables), while in the MLIP method it is obtained from matrix-based approach. The second-order polynomial by both methods is closer by applying the direct MLS method that does not guarantee even if the values are very small or the determinant is close to zero. If the matrix values are too small, we cannot determine the inverse and its product of matrices. Thus, the matrix of these methods depends on the size: the matrix column as the number of variables with same size of sample runs constructs a square matrix. Different case studies were discussed for fitting the linear and second-order polynomials over number of independent versus dependent variables. Thus, the MLIP method has less computation with less time consumption. For both applications, the MLIP method is a more adoptable easy computation for solving real-time applications. As a future work, this approach is extended for the quantification of reliability in the second problem. The number of variables is greater than the number of sample runs (this matrix must be rectangular), then a new procedure is required for polynomial computations. -195