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Review—Fortran

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Review—I/O

Patterns:

Read until a sentinel value is found

Read n, then read n things

Read until EOF encountered

Techniques:

Unformatted reads (preferred)

Formatted reads (deprecated)

Unformatted writes (deprecated)

Formatted writes (preferred)

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Conditional Execution

IF (…) THEN

…

ELSE

…

END IF

Keep it simple

Follow rules of boolean logic

Use Logical variables if necessary

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Loops

DO n = begin,end,step

END DO

…………………………………………………..

DO

IF(we’re done) THEN

EXIT

END IF

…

END DO

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Loops

X = initial value

DO n = begin,end,step

update X

END DO

Works for finding min, max,

Works for sum of values, product of values

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Loops

DO

IF(we’re done) THEN

EXIT

END IF

…

END DO

Works for searching, indeterminate termination conditions

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Data Types and Variables

Intrinsic

logical, character, integer, real, complex

Structures

Static or dynamic allocation

Call by reference, call by value

INTENT keyword

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Subprograms and Scope of Variables

Functions and subroutines

Internal and external

Variables are available to internal subprograms unless otherwise redefined

Variables are not available to external subprograms

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Algorithms

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Pointers to Data

Sorting with an index array

Linked lists, queues, stacks

Old fashioned way used arrays• bad—more cumbersome?• good—all the data can be read

More modern way uses dynamic allocation• bad—can’t necessarily find the data• good—perhaps simpler

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Divide and Conquer

Bisection is divide and conquer

Split the problem in half

Determine which half is relevant

Inherently fast

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Algorithm Analysis

f(n) = O(g(x)) means f(x) < C g(x) for some fixed constant C and “all large x”

If f(x) is a polynomial, then it’s big O of the leading term

Hierarchy

log n

n

n log n

n^2

…

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Numerical Algorithms

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Root Finding

Bisection method

more or less guaranteed to work

but slow to converge

Newton’s method

generally fast in convergence

but less predictable

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Approximation of Functions

Forward (backward) difference approximations:• Approximate derivatives with slopes from data

points forward (backward) from an initial base pointCentral difference approximations:• Approx with average forward and backwardBoth methods approx a function with a polynomial by

approximating the derivatives using data points and then approximating the function with a Taylor series

Lagrange interpolation:Create the exact polynomial passing through the data

points.

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Approximation of Functions

When the data is felt to be accurate, difference methods are appropriate.

When the data is known to be inaccurate, a least squares approximation is used.

Usually: Minimize the sums of the errors in the vertical (y) direction if we were to approx the function with a straight line

If we have nonlinear data, do a transformation, then approximate with a straight line.

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Means, Variances, etc.

Mean can be computed “online”

Variance can be computed online, but is usually expressed as a computation that requires first computing the mean

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Random Numbers

Monte Carlo computations

Numerical integration

Simulation of statistical events (traffic flow, particle interactions, etc.)

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Differential Equations

Start from an initial condition (time zero, for example)

Estimate rate of change numerically

Proceed in the direction of that change for one time step

Recalculate rates of changes

repeat

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Sorting

Bubble sort/insertion sort, O(n^2) time, but there are faster ways

Sort data?

Sort keys?

Create an index array but don’t move the data itself?

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Matrix Operations

Matrix multiplication

Loops

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Gaussian Elimination

The most basic of all direct linear system solvers

Also not very stable or successful, and not very efficient

Best done as vector-matrix multiplication either explicitly or implicitly

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Gauss-Seidel Iteration

The most basic of all iterative linear system solvers

Reasonably stable and successful, and reasonably efficient

Can be done either sequentially (G-S) or in parallel (Jacobi iteration)

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The End