learning matlab problem solving and numerical analysis through examples

Numerical Methods in MATLAB

• First Online: 08 March 2022

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• Eklas Hossain 2  

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Numerical analysis is useful to solve numerous engineering problems encompassing several applications. Although there are many diverse and complicated branches of numerical analysis, only three important and reader-friendly methods are included in this chapter. This chapter aims to deliver the basics of the most commonly deployed numerical methods with worked out examples in MATLAB so that the readers get acquainted with the fundamental approach of numerical analysis. This concise chapter describes the Gauss-Seidel method, the Newton-Raphson method, and the Runge-Kutta method elaborately along with MATLAB examples for each method. Each of these three methods is first addressed theoretically with the mathematical equations, and then the MATLAB examples are suitably designed to help the readers understand how to solve different types of equations and also determine the tolerance of the solutions. Having completed this chapter, a reader will be able to solve equations using numerical methods such as the Gauss-Seidel, Newton-Raphson, and Runge-Kutta methods.

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Oregon Institute of Technology, Klamath Falls, OR, USA

Eklas Hossain

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Correspondence to Eklas Hossain .

Write down the basic steps of the following:

Gauss-Seidel method

Newton-Raphson method

Runge-Kutta method

State the major differences in the calculation of the three methods mentioned in Question 1.

Given a set of equation as follows:

Solve the equation using Gauss-Seidel method in MATLAB. Consider the tolerance for x , y , and  z to be less than 0.00001.

What happens when you decrease the tolerance to 0.0001 and 0.001? Do the values vary from the values determined in question (a)? Why do you think this occurs?

What happens when you increase the tolerance to 0.000001 and 0.0000001? Do the values vary from the values determined in question (a)? Why do you think this occurs?

Solve the following equation using Newton-Raphson method, which has a root within the range of [0, 2]. Consider a tolerance for the value of root less than 0.0001:

3 x  + 2 cos ( x ) − 5

x 5  −  x  − 2

Use the classical fourth-order Runge-Kutta method to solve the following differential equations for the step size of 0.2, for 0 ≤  x  ≤ 2, and with an initial condition of y (0) = 5 in MATLAB:

\( \frac{dy}{dx}=-4{x}^3-6{x}^2-10x+2 \)

\( \frac{dy}{dx}= xsin(y)+y\cos (x) \)

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About this chapter

Hossain, E. (2022). Numerical Methods in MATLAB. In: MATLAB and Simulink Crash Course for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-89762-8_7

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Published : 08 March 2022

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Train Network with Numeric Features

This example shows how to create and train a simple neural network for deep learning feature data classification.

If you have a data set of numeric features (for example a collection of numeric data without spatial or time dimensions), then you can train a deep learning network using a feature input layer. For an example showing how to train a network for image classification, see Create Simple Deep Learning Neural Network for Classification .

This example shows how to train a network to classify the gear tooth condition of a transmission system given a mixture of numeric sensor readings, statistics, and categorical labels.

Load the transmission casing dataset for training. The data set consists of 208 synthetic readings of a transmission system consisting of 18 numeric readings and three categorical labels:

SigMean — Vibration signal mean

SigMedian — Vibration signal median

SigRMS — Vibration signal RMS

SigVar — Vibration signal variance

SigPeak — Vibration signal peak

SigPeak2Peak — Vibration signal peak to peak

SigSkewness — Vibration signal skewness

SigKurtosis — Vibration signal kurtosis

SigCrestFactor — Vibration signal crest factor

SigMAD — Vibration signal MAD

SigRangeCumSum — Vibration signal range cumulative sum

SigCorrDimension — Vibration signal correlation dimension

SigApproxEntropy — Vibration signal approximate entropy

SigLyapExponent — Vibration signal Lyap exponent

PeakFreq — Peak frequency.

HighFreqPower — High frequency power

EnvPower — Environment power

PeakSpecKurtosis — Peak frequency of spectral kurtosis

SensorCondition — Condition of sensor, specified as "Sensor Drift" or "No Sensor Drift"

ShaftCondition — Condition of shaft, specified as "Shaft Wear" or "No Shaft Wear"

GearToothCondition — Condition of gear teeth, specified as "Tooth Fault" or "No Tooth Fault"

Read the transmission casing data from the CSV file "transmissionCasingData.csv" .

Convert the labels for prediction to categorical using the convertvars function.

View the first few rows of the table.

To train a network using categorical features, you must first convert the categorical features to numeric. First, convert the categorical predictors to categorical using the convertvars function by specifying a string array containing the names of all the categorical input variables. In this data set, there are two categorical features with names "SensorCondition" and "ShaftCondition" .

Loop over the categorical input variables. For each variable:

Convert the categorical values to one-hot encoded vectors using the onehotencode function.

Add the one-hot vectors to the table using the addvars function. Specify to insert the vectors after the column containing the corresponding categorical data.

Remove the corresponding column containing the categorical data.

Split the vectors into separate columns using the splitvars function.

View the first few rows of the table. Notice that the categorical predictors have been split into multiple columns with the categorical values as the variable names.

View the class names of the data set.

Split Data Set into Training and Validation Sets

Partition the data set into training, validation, and test partitions. Set aside 15% of the data for validation, and 15% for testing.

View the number of observations in the dataset.

Determine the number of observations for each partition.

Create an array of random indices corresponding to the observations and partition it using the partition sizes.

Partition the table of data into training, validation, and testing partitions using the indices.

Define Network Architecture

Define the network for classification.

Define a network with a feature input layer and specify the number of features. Also, configure the input layer to normalize the data using Z-score normalization. Next, include a fully connected layer with output size 50 followed by a batch normalization layer and a ReLU layer. For classification, specify another fully connected layer with output size corresponding to the number of classes, followed by a softmax layer.

Specify Training Options

Specify the training options.

Train the network using Adam.

Train using mini-batches of size 16.

Shuffle the data every epoch.

Monitor the network accuracy during training by specifying validation data.

Display the training progress in a plot and suppress the verbose command window output.

The software trains the network on the training data and calculates the accuracy on the validation data at regular intervals during training. The validation data is not used to update the network weights.

Train Network

Train the network using the architecture defined by layers , the training data, and the training options. By default, trainnet uses a GPU if one is available, otherwise, it uses a CPU. Training on a GPU requires Parallel Computing Toolbox™ and a supported GPU device. For information on supported devices, see GPU Computing Requirements (Parallel Computing Toolbox) . You can also specify the execution environment by using the ExecutionEnvironment name-value argument of trainingOptions .

The training progress plot shows the mini-batch loss and accuracy and the validation loss and accuracy. For more information on the training progress plot, see Monitor Deep Learning Training Progress .

Test Network

Predict the labels of the test data using the trained network and calculate the accuracy. Specify the same mini-batch size used for training.

Calculate the classification accuracy. The accuracy is the proportion of the labels that the network predicts correctly.

View the results in a confusion matrix.

trainnet | trainingOptions | dlnetwork | fullyConnectedLayer | Deep Network Designer | featureInputLayer

Related Examples

• Create Simple Deep Learning Neural Network for Classification
• Train Convolutional Neural Network for Regression
• Deep Learning in MATLAB
• List of Deep Learning Layers

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