Scientific Calculator

Regression Analysis with the Calculator

This calculator is designed for simple and multiple regression analysis. The mathematical relationship between the independent variables X₁, X₂, and X₃ and the dependent variable Y is estimated using the least-squares method. One, two, or three independent variables can be used simultaneously, and different mathematical transformations can be selected for each variable.

Supported Models

The following models can be selected independently for each independent variable (X₁, X₂, and X₃):

• Linear: x
• Polynomial: x + x², x + x² + x³
• Square root: √x
• Reciprocal functions: 1/x, 1/x + 1/x²
• Logarithmic: ln(x), log₁₀(x)
• Exponential: exp(x), 10^x

The following transformations can be applied to the dependent variable (Y):

• y
• ln(y)
• log₁₀(y)
• e^y
• 10^y
• y²
• y³

Data Input

Enter the data in the input field using one of the following formats:

x1,y
x1,x2,y
x1,x2,x3,y

When entering multiple data rows, rows must be separated by either a ; character or a new line (Enter). Do not use both separators within the same data set.

Data copied from Excel can be pasted directly into the input field. Line breaks are automatically interpreted as separate data rows when necessary.

Regression Calculation

Use the R+ button to add data. If necessary, an existing record in the history can be edited or deleted. Press the = button to calculate the regression model.

The calculator provides the following statistical results:

• Regression equation
• Coefficients (a, b₁, b₂, ...)
• R² (Coefficient of Determination)
• Adjusted R²
• RMSE (Root Mean Square Error)
• MAE (Mean Absolute Error)
• Maximum error
• Predicted value (ŷ) and error for each data row

Prediction (x→y)

After the regression model has been calculated, enter new X values in the input field and press the x→y button. The calculator evaluates the selected model and displays the predicted Y value in the input field.

Example Data

1,1,10;
2,1,12;
3,1,14;
1,2,13;
2,2,15;
3,2,17;
1,3,16;
2,3,18;
3,3,20;
4,4,25;
5,5,30;
6,6,35;

For this example data set, selecting the linear model for X₁ and X₂ produces the following equation:

y = 5 + 2·x₁ + 3·x₂

Because this data set was constructed to follow the equation exactly, R² = 1, RMSE = 0, and MAE = 0. This provides a simple way to verify that the regression engine is working correctly.

Solving Systems of Equations with the Calculator

This calculator provides numerical solutions for systems of linear and nonlinear equations with multiple unknowns. The number of equations must be equal to the number of unknowns.

1. Defining Variables

First, enter the unknown variables used in the system, separated by commas, and press Enter.

x,y,z
x1,x2,x3,y1,y2

Sequentially numbered variables can also be defined using a shortened notation:

x1..x10
P1..P20,Q1..Q20

A variable name cannot be defined more than once.

2. Initial Values

The default initial value for all variables is 1. Initial values do not need to be entered separately.

To use a different initial value, select the variable from the Initial Value list. The selected variable will appear in the input field, for example, as x1=1.

x1=5
P1=2.5

Setting the value back to 1 restores the default initial value. In nonlinear systems, initial values may affect convergence and the solution obtained.

3. Defining Constants

Constants are optional. Multiple constants can be defined at once by separating them with commas.

K=80,C=120,L=25

A constant cannot have the same name as a defined variable. If no constants are required, leave the input field empty and press Enter.

4. Defining Custom Functions

Repeated mathematical expressions can be defined as custom functions. This step is optional.

flow(p)=K*sqrt(p)
f(x)=x^2+2*x+1
loss(q,L)=8.89e-6*L*q^1.85

Function parameters are local variables used only within the function. Previously defined constants can be used directly inside custom functions.

For example, if the constant K=80 has been defined:

flow(p)=K*sqrt(p)

the function can then be used in equations as:

q4=flow(p4)

Multiple custom functions can be entered separately. When all function definitions are complete, leave the input field empty and press Enter to proceed to equation entry.

A custom function in the history can be clicked to edit and update it.

5. Entering Equations

Enter the equations one at a time in sequence. Equations can be entered using standard mathematical notation.

q4=K*sqrt(p4)
x^2+y^2=25
x-y=1

The calculator automatically converts the left-hand and right-hand sides of an equation into the form F(X)=0.

For example:

x^2+y^2=25

is internally evaluated as:

(x^2+y^2)-(25)=0

An expression already written in F(X)=0 form can also be entered without an equals sign:

x^2+y^2-25

Each equation is added to the history by pressing Enter. An equation in the history can be clicked to edit and update it.

When the number of equations reaches the number of unknowns, the following message is displayed:

20 / 20 equations completed · Ready to solve

The system can then be solved using the = button.

6. Supported Mathematical Functions

sin(x), cos(x), tan(x)
asin(x), acos(x), atan(x)
sinh(x), cosh(x), tanh(x)
sqrt(x), cbrt(x)
ln(x), log(x), log10(x)
exp(x), pow(x,y)
abs(x), min(a,b), max(a,b)

The mathematical constants pi and e can also be used. Use the ^ character for exponentiation.

The System of Equations mode is intended for systems with real-valued solutions.

7. Solution Method

The system of equations is solved iteratively using the Newton-Raphson method. The Jacobian matrix is calculated using numerical derivatives. When necessary, the Newton step is reduced to improve convergence.

Nonlinear systems may have more than one mathematical solution. In such cases, changing the initial values may lead to different solutions.

8. Results and Equation Residuals

After the calculation is completed, the calculated value of each unknown and the residual of each equation at the solution point are displayed in the history.

x = 4        F1 = 0
y = 3        F2 = 0

F1, F2, ... values equal to or very close to zero indicate that the calculated variable values satisfy the corresponding equations.

The number of iterations and the maximum equation residual, Max |F(X)|, are also displayed.

9. Saving and Loading a Calculation

System of Equations calculations can be saved as .kilicaslan working files using the Save button.

The file stores the following information:

• Variables
• All initial values
• Constants
• Custom functions
• Equations
• Calculated variable values
• Equation residuals
• Iteration and convergence information

A previously saved .kilicaslan file can be opened using the Load button. The file format, module, and version information are checked during loading.

The loaded variables, constants, custom functions, equations, and calculation results, if available, are restored in the history.

10. Example — Sprinkler Flow and Pressure Equations

Variables:

q4,p4

Constant:

K=80

Custom function:

flow(p)=K*sqrt(p)

Press Enter with an empty input field to finish entering custom functions. Then enter the equations:

q4=flow(p4)
p4+8.89e-6*q4^1.85=0.752

Since there are two variables and two equations, the system is ready to be solved. Press the = button to calculate q4 and p4.

11. Input Sequence

In System of Equations mode, data is entered in the following order:

Variables
↓
Constants (press Enter if none)
↓
Custom Functions (press Enter if none)
↓
Equations
↓
=
↓
Solution

Vector Operations with the Calculator

This calculator can perform basic and advanced operations on two-dimensional and three-dimensional vectors. Vectors are entered using square brackets. For example, [3,4] is a two-dimensional vector, while [1,2,3] is a three-dimensional vector.

Vector Input

[3,4]
[1,2,3]
[5,-2,7]

Vector components are separated by commas. Addition, subtraction, scalar multiplication, and division can be performed directly.

[1,2] + [3,4]
[5,7] - [2,3]
[3,4] * 2
[6,8] / 2

Vector Magnitude and Unit Vector

The magnitude of a vector can be calculated using the norm() function or the ‖ ‖ notation.

norm([3,4])
‖[3,4]‖

For the vector [3,4], the result is 5.

To calculate a unit vector, use the unit() function or the û symbol.

unit([3,4])
û[3,4]

The result is approximately [0.6, 0.8].

Dot Product

The dot product measures how strongly two vectors point in the same direction. Use the dot() function or the • symbol.

dot([1,2],[3,4])
[1,2] • [3,4]

The result in this example is 11.

Cross Product

For three-dimensional vectors, the cross product can be calculated using the cross() function or the × symbol.

cross([1,0,0],[0,1,0])
[1,0,0] × [0,1,0]

The result is [0,0,1]. This vector indicates a direction perpendicular to both input vectors.

Angle Between Vectors

The angle between two vectors can be calculated using the ∠ symbol. The result is displayed in DEG, RAD, or GRAD according to the selected angle mode.

[1,0] ∠ [0,1]

In DEG mode, the result is 90.

Vector Rotation

A two-dimensional vector can be rotated through a specified angle.

rot([1,0],90)

In DEG mode, the result is approximately [0,1].

The rot() function can also be used to rotate one vector toward another vector.

rot([1,0],[0,1])
rot([1,0],[0,1],30)

With two arguments, the first vector is rotated to align with the direction of the second vector. With three arguments, the first vector is rotated toward the second vector by the specified angle.

Projection

The projection of one vector onto another can be calculated using the proj() function.

proj([3,4],[1,0])

In this example, the projection of [3,4] onto the x-axis is [3,0].

Example Usage

‖[3,4]‖
[1,2] • [3,4]
[1,0] ∠ [0,1]
rot([1,0],90)
proj([3,4],[1,0])

These examples can be used to test vector magnitude, dot product, angle calculation, rotation, and projection operations.