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Numerical Algs. Inner products, Matrix Multiplication, LU-Decomposition (factorization) Simplex Method, Iterative methods ( sqrt ). Optimization. Many problems in computing are of the form: Find the values that minimize (or maximizes) some function subject to constraints.
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Numerical Algs Inner products, Matrix Multiplication, LU-Decomposition (factorization) Simplex Method, Iterative methods (sqrt)
Optimization • Many problems in computing are of the form: • Find the values that minimize (or maximizes) some function subject to constraints
Example (unconstrained) • Find mean and standard deviation of a normal distribution that maximizes the probability of sampleswhere the probability is
Example (unconstrained) • Find a quadratic polynomial where we have made measurementsso that the mean squared error in our measurements is minimized.
Example • (Help me out here)
Matrices • This REALLY should be review for you…. • A matrix is a 2D array of numbers (real or complex) • The number of rows is usually called • The number of columns is usually called • Matrices are represented by upper case letters • Matrix elements two subscripts • is the element in row and column of matrix • Indices start at 1, not zero • Sometimes elements are written lowercase, sometime uppercase. (sloppy) • Matrix multiplication is special
Matrix + Matrix • Addition/Subtraction is done element by element
Matrix * Scalar • Done in parallel • Symmetric ()
Matrix Transpose • Use superscript to indicate transpose. • Exchange the order of subscripts (rows become columns)
Slicing • In this class I will use slicing to indicate a sub-array. • My notation is something like Matlab’s. • Concatenation • I will use the | operator to concatenate matrices: , , or
Vectors • A vector is a single-column matrix. • We write them with lowercase boldface letters. • A row-vector is a single-row matrix. We always write them as to indicate they are the transpose of a column matrix.
Inner Product • Defined for two vectors Other notations for inner product: , , Also called the ‘dot’ product.
Types of Matrix • Zero • Identity • Symmetric • Lower Triangular • Strict Lower Triangular • Upper & Strict Upper • Banded
Accessing Matrix Elements If we know the dimensions of the array ahead of time we can use a fixed-size 2D array. But usually the dimensions are decided at runtime. I cannot pass this as a parameter to a function that operates on arrays with a variable size
Writing functions for 2D arrays • We need to do some extra work to be able to pass matrices arount
Accessing Matrix Elements • The layout is ‘contiguous’ in memory. • The is row major order. • The difference between items on the same row and different columns is _______ • The difference between items on the same column and different rows is _______ • The index of the item at row and column is ________
Accessing Matrix Elements • Other API’s use different layout (Fortran, OpenGL, MatLAB, …) • This is column major order. • The difference between items on the same row and different columns is _______ • The difference between items on the same column and different rows is _______ • The index of the item at row and column is ________ • LaPACK is the most commonly used library for math. • It is implemented in Fortran, and called from c / c++.
Representing a slice (MatLAB style) Let denote the numbers So Consider int A[] = Row 1 of has indices 0:1:2. Column 1 of A has indices 0:3:9.
Basic idea of a slice • Keep track of the start index (or pointer) • Keep track of the number of values (or end index) • Keep track of the difference between consecutive values (stepsize)
C++ Slice • std::slice • Start • Size (number of items) • Step (difference between consecutive items) • std::slice(0, 3, 2) => 0, 2, 4 Valarray<double> x; ….. x[slice(0,3,2)] => x[0], x[2], x[4] • I don’t use std::slice often (I write my own matrix class or use another library) • The idea is very important
Matrix Views & Submatrices • What if we want a submatrix that is not a single row or column? How can we represent ? • We need a 2D version of a slice • Keep track of… • Linear index of in this case • Offset between consecutive rows is , in this case 3 • Offset between consecutive columns is • Keep track of the sizes (in this case )
Matrix View • Imagine that library code & functions always operate on views of some other underlying data. • A vector view might need the following: • A pointer to the memory • Offset to the first item • Offset between consecutive items • The number of items in the vector. • A matrix view might need the following: • Pointer to the memory • Offset to the first item • Offset between consecutive rows • Offset between consecutive items • Number of rows • Number of columns In the STL these are called std::slice_array but do not seem to be used often. There are many custom implementations of this idea though. In the STL these are called std::gslice_array but do not seem to be used often. There are many custom implementations of this idea though.
Matrix Multiplication If then Dimensions must agree: The number of columns in A must match the number of rows in B
