## Spline Approximation (Mathematics II)

This is the third of a series of article about spline approximation. If you have not done so, you should start reading Spline Approximation (Introduction) Spline Approximation (Mathematics) So a function which is supposed to approximate a given set of points as a linear combination     of functions , as described in the previous …

## Spline Approximation (Mathematics)

The goal of spline approximation has already been explained in the previous article „Spline Approximation (Introduction)„. This article will cover the mathematics behind this approximation and develop an approach. If you do not care about the mathematics, just skip this article and read the Spline Approximation (Cookbook)“, that will come soon. Spline Interpolation We have …

## Spline Approximation (Introduction)

We sometimes encounter a situation where a number of points with coordinates are given and we want to find a function such that for all of these points we have (interpolation) or (approximation). Most often we say that we want on average to be as small as possible and for whatever reasons usually the quadratic …

## GIMP

In spite of working mostly for server software and server setup using powerful non-graphic command line tools and scripting languages, it is sometimes fun to work with something very graphical. I did talk about Clojure Art, which is fun and creates interesting visual results and helps getting into the phantastic language Clojure. But more than …

## Combining multiple scans

When images are scanned multiple times, maybe there is a way to construct an image that is better than any of the scans from them. In this case it is assumed, that one scan has a higher resolution, but another scan got the colors better. It has already been found out, which two scans belong …

## Ranges of Dates and Times

In Software we often deal with ranges of dates and times. Let us look at it from the perspective of an end user. When we say something like „from 2020-03-07 to 2019-03-10“ we mean the set of all timestamps such that     or more accurately:     Important is, that we mean to include …

## How to recover the Borrow Bit

In a similar way as the carry bit for addition it is possible to recover the borrow bit for substraction, just based on the highest bits of three numbers that we deal with during the operation. With this program, a subtraction operation of an 8-bit CPU can be simulated exhaustively #!/usr/bin/perl my \$x, \$x, \$bi; my %tab = (); for (\$bi = 0; \$bi  7;             my \$zs = \$z >> 7; …

## Borrow and Carry Bit for Subtraction

Similar to the usage of the carry bit when adding there are mechanisms for subtracting that allow to integrate the result of subtraction of the lower bits into the subtraction of the next higher block of bits, where necessary. There are two ways to do this, that are trivially equivalent by a simple not operation: …

## Rounding to Rational Numbers

Usually we think of rounding as a way of approximately expressing numbers with many decimal places by numbers with fewer decimal places. Regular readers of this blog have already encountered, that this concept can be extended and generalized, as is mentioned in the article about Geometric and Harmonic Rounding and Residue Class Rounding or Rounding …

## How to calculate Square Roots and Cubic Roots

The functions sqrt and sometimes even cbrt are commonly available, but it is nice to see how they can be calculated. There are several approaches, but the most popular ones are Newton’s method and an algorithmic formulation of how roots are taken manually, for those old enough to still have learned it in school. Earlier …