Microsoft Excel

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Microsoft Excel

A simple bar graph being created in Excel, running on Windows 11
Developer(s)	Microsoft
Initial release	November 19, 1987; 35 years ago
Stable release	2309 (16827.20130) / September 28, 2023; 33 days ago
Written in	C++ (back-end)
Operating system	Microsoft Windows
Type	Spreadsheet
License	Trialware
Website	microsoft.com/en-us/microsoft-365/excel

Microsoft Excel for Mac

Excel for Mac (version 16.67), running on macOS Big Sur 11.5.2
Developer(s)	Microsoft
Initial release	September 30, 1985
Stable release	16.70 (Build 23021201) / February 14, 2023; 8 months ago
Written in	C++ (back-end), Objective-C (API/UI)
Operating system	macOS
Type	Spreadsheet
License	Proprietary commercial software
Website	products.office.com/mac

Microsoft Excel for Android

Excel for Android running on Android 13
Developer(s)	Microsoft Corporation
Stable release	16.0.16501.20160 / May 26, 2023; 5 months ago
Operating system	Android Pie or later
Type	Spreadsheet
License	Proprietary commercial software
Website	products.office.com/en-us/excel

Microsoft Excel for iOS and iPadOS

Developer(s)	Microsoft Corporation
Stable release	2.73 / May 15, 2023; 5 months ago
Operating system	iOS 15 or later iPadOS 15 or later
Type	Spreadsheet
License	Proprietary commercial software
Website	products.office.com/en-us/excel

Microsoft Excel is a spreadsheet editor developed by Microsoft for Windows, macOS, Android, iOS and iPadOS. It features calculation or computation capabilities, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications (VBA). Excel forms part of the Microsoft 365 suite of software.

Features

Basic operation

Main article: Spreadsheet

Microsoft Excel has the basic features of all spreadsheets, using a grid of cells arranged in numbered rows and letter-named columns to organize data manipulations like arithmetic operations. It has a battery of supplied functions to answer statistical, engineering, and financial needs. In addition, it can display data as line graphs, histograms and charts, and with a very limited three-dimensional graphical display. It allows sectioning of data to view its dependencies on various factors for different perspectives (using pivot tables and the scenario manager). A PivotTable is a tool for data analysis. It does this by simplifying large data sets via PivotTable fields. It has a programming aspect, Visual Basic for Applications, allowing the user to employ a wide variety of numerical methods, for example, for solving differential equations of mathematical physics, and then reporting the results back to the spreadsheet. It also has a variety of interactive features allowing user interfaces that can completely hide the spreadsheet from the user, so the spreadsheet presents itself as a so-called application, or decision support system (DSS), via a custom-designed user interface, for example, a stock analyzer, or in general, as a design tool that asks the user questions and provides answers and reports. In a more elaborate realization, an Excel application can automatically poll external databases and measuring instruments using an update schedule, analyze the results, make a Word report or PowerPoint slide show, and e-mail these presentations on a regular basis to a list of participants. Excel was not designed to be used as a database.

Microsoft allows for a number of optional command-line switches to control the manner in which Excel starts.

Functions

See also: Microsoft Power Fx

Excel 2016 has 484 functions. Of these, 360 existed prior to Excel 2010. Microsoft classifies these functions into 14 categories. Of the 484 current functions, 386 may be called from VBA as methods of the object "WorksheetFunction" and 44 have the same names as VBA functions.

With the introduction of LAMBDA, Excel became Turing complete.

Macro programming

VBA programming

Main article: Visual Basic for Applications

Use of a user-defined function sq(x) in Microsoft Excel. The named variables x & y are identified in the Name Manager. The function sq is introduced using the Visual Basic editor supplied with Excel.

Subroutine in Excel calculates the square of named column variable x read from the spreadsheet, and writes it into the named column variable y.

The Windows version of Excel supports programming through Microsoft's Visual Basic for Applications (VBA), which is a dialect of Visual Basic. Programming with VBA allows spreadsheet manipulation that is awkward or impossible with standard spreadsheet techniques. Programmers may write code directly using the Visual Basic Editor (VBE), which includes a window for writing code, debugging code, and code module organization environment. The user can implement numerical methods as well as automating tasks such as formatting or data organization in VBA and guide the calculation using any desired intermediate results reported back to the spreadsheet.

VBA was removed from Mac Excel 2008, as the developers did not believe that a timely release would allow porting the VBA engine natively to Mac OS X. VBA was restored in the next version, Mac Excel 2011, although the build lacks support for ActiveX objects, impacting some high level developer tools.

A common and easy way to generate VBA code is by using the Macro Recorder. The Macro Recorder records actions of the user and generates VBA code in the form of a macro. These actions can then be repeated automatically by running the macro. The macros can also be linked to different trigger types like keyboard shortcuts, a command button or a graphic. The actions in the macro can be executed from these trigger types or from the generic toolbar options. The VBA code of the macro can also be edited in the VBE. Certain features such as loop functions and screen prompt by their own properties, and some graphical display items, cannot be recorded but must be entered into the VBA module directly by the programmer. Advanced users can employ user prompts to create an interactive program, or react to events such as sheets being loaded or changed.

Macro Recorded code may not be compatible with Excel versions. Some code that is used in Excel 2010 cannot be used in Excel 2003. Making a Macro that changes the cell colors and making changes to other aspects of cells may not be backward compatible.

VBA code interacts with the spreadsheet through the Excel Object Model, a vocabulary identifying spreadsheet objects, and a set of supplied functions or methods that enable reading and writing to the spreadsheet and interaction with its users (for example, through custom toolbars or command bars and message boxes). User-created VBA subroutines execute these actions and operate like macros generated using the macro recorder, but are more flexible and efficient.

History

From its first version Excel supported end-user programming of macros (automation of repetitive tasks) and user-defined functions (extension of Excel's built-in function library). In early versions of Excel, these programs were written in a macro language whose statements had formula syntax and resided in the cells of special-purpose macro sheets (stored with file extension .XLM in Windows.) XLM was the default macro language for Excel through Excel 4.0. Beginning with version 5.0 Excel recorded macros in VBA by default but with version 5.0 XLM recording was still allowed as an option. After version 5.0 that option was discontinued. All versions of Excel, including Excel 2021 are capable of running an XLM macro, though Microsoft discourages their use.

Python programming

In 2023 Microsoft announced Excel would support the Python programming language directly.

Charts

Graph made using Microsoft Excel

Excel supports charts, graphs, or histograms generated from specified groups of cells. It also supports Pivot Charts that allow for a chart to be linked directly to a Pivot table. This allows the chart to be refreshed with the Pivot Table. The generated graphic component can either be embedded within the current sheet or added as a separate object.

These displays are dynamically updated if the content of cells changes. For example, suppose that the important design requirements are displayed visually; then, in response to a user's change in trial values for parameters, the curves describing the design change shape, and their points of intersection shift, assisting the selection of the best design.

Add-ins

Additional features are available using add-ins. Several are provided with Excel, including:

• Analysis ToolPak: Provides data analysis tools for statistical and engineering analysis (includes analysis of variance and regression analysis)
• Analysis ToolPak VBA: VBA functions for Analysis ToolPak
• Euro Currency Tools: Conversion and formatting for euro currency
• Solver Add-In: Tools for optimization and equation solving

Data storage and communication

Number of rows and columns

Versions of Excel up to 7.0 had a limitation in the size of their data sets of 16K (2¹⁴ = 16384) rows. Versions 8.0 through 11.0 could handle 64K (2¹⁶ = 65536) rows and 256 columns (2⁸ as label 'IV'). Version 12.0 onwards, including the current Version 16.x, can handle over 1M (2²⁰ = 1048576) rows, and 16384 (2¹⁴, labeled as column 'XFD') columns.

File formats

Excel Spreadsheet

Filename extension	.xls, (.xlsx, .xlsm, .xlsb - Excel 2007)
Internet media type	application/vnd.ms-excel
Uniform Type Identifier (UTI)	com.microsoft.excel.xls
Developed by	Microsoft
Type of format	Spreadsheet

Microsoft Excel up until 2007 version used a proprietary binary file format called Excel Binary File Format (.XLS) as its primary format. Excel 2007 uses Office Open XML as its primary file format, an XML-based format that followed after a previous XML-based format called "XML Spreadsheet" ("XMLSS"), first introduced in Excel 2002.

Although supporting and encouraging the use of new XML-based formats as replacements, Excel 2007 remained backwards-compatible with the traditional, binary formats. In addition, most versions of Microsoft Excel can read CSV, DBF, SYLK, DIF, and other legacy formats. Support for some older file formats was removed in Excel 2007. The file formats were mainly from DOS-based programs.

Binary

OpenOffice.org has created documentation of the Excel format. Two epochs of the format exist: the 97-2003 OLE format, and the older stream format. Microsoft has made the Excel binary format specification available to freely download.

XML Spreadsheet

Main article: Microsoft Office XML formats

The XML Spreadsheet format introduced in Excel 2002 is a simple, XML based format missing some more advanced features like storage of VBA macros. Though the intended file extension for this format is .xml, the program also correctly handles XML files with .xls extension. This feature is widely used by third-party applications (e.g. MySQL Query Browser) to offer "export to Excel" capabilities without implementing binary file format. The following example will be correctly opened by Excel if saved either as Book1.xml or Book1.xls:

<?xml version="1.0"?>
<Workbook xmlns="urn:schemas-microsoft-com:office:spreadsheet"
 xmlns:o="urn:schemas-microsoft-com:office:office"
 xmlns:x="urn:schemas-microsoft-com:office:excel"
 xmlns:ss="urn:schemas-microsoft-com:office:spreadsheet"
 xmlns:html="http://www.w3.org/TR/REC-html40">
 <Worksheet ss:Name="Sheet1">
  <Table ss:ExpandedColumnCount="2" ss:ExpandedRowCount="2" x:FullColumns="1" x:FullRows="1">
   <Row>
    <Cell><Data ss:Type="String">Name</Data></Cell>
    <Cell><Data ss:Type="String">Example</Data></Cell>
   </Row>
   <Row>
    <Cell><Data ss:Type="String">Value</Data></Cell>
    <Cell><Data ss:Type="Number">123</Data></Cell>
   </Row>
  </Table>
 </Worksheet>
</Workbook>

Current file extensions

Main article: Office Open XML

Microsoft Excel 2007, along with the other products in the Microsoft Office 2007 suite, introduced new file formats. The first of these (.xlsx) is defined in the Office Open XML (OOXML) specification.

Excel 2007 formats

Format	Extension	Description
Excel Workbook	.xlsx	The default Excel 2007 and later workbook format. In reality, a ZIP compressed archive with a directory structure of XML text documents. Functions as the primary replacement for the former binary .xls format, although it does not support Excel macros for security reasons. Saving as .xlsx offers file size reduction over .xls
Excel Macro-enabled Workbook	.xlsm	As Excel Workbook, but with macro support.
Excel Binary Workbook	.xlsb	As Excel Macro-enabled Workbook, but storing information in binary form rather than XML documents for opening and saving documents more quickly and efficiently. Intended especially for very large documents with tens of thousands of rows, and/or several hundreds of columns. This format is very useful for shrinking large Excel files as is often the case when doing data analysis.
Excel Macro-enabled Template	.xltm	A template document that forms a basis for actual workbooks, with macro support. The replacement for the old .xlt format.
Excel Add-in	.xlam	Excel add-in to add extra functionality and tools. Inherent macro support because of the file purpose.

Old file extensions

Format	Extension	Description
Spreadsheet	.xls	Main spreadsheet format which holds data in worksheets, charts, and macros
Add-in (VBA)	.xla	Adds custom functionality; written in VBA
Toolbar	.xlb	The file extension where Microsoft Excel custom toolbar settings are stored.
Chart	.xlc	A chart created with data from a Microsoft Excel spreadsheet that only saves the chart. To save the chart and spreadsheet save as .XLS. XLC is not supported in Excel 2007 or in any newer versions of Excel.
Dialog	.xld	Used in older versions of Excel.
Archive	.xlk	A backup of an Excel Spreadsheet
Add-in (DLL)	.xll	Adds custom functionality; written in C++/C, Fortran, etc. and compiled into a special dynamic-link library
Macro	.xlm	A macro is created by the user or pre-installed with Excel.
Template	.xlt	A pre-formatted spreadsheet created by the user or by Microsoft Excel.
Module	.xlv	A module is written in VBA (Visual Basic for Applications) for Microsoft Excel
Library	.DLL	Code written in VBA may access functions in a DLL, typically this is used to access the Windows API
Workspace	.xlw	Arrangement of the windows of multiple Workbooks

Using other Windows applications

Windows applications such as Microsoft Access and Microsoft Word, as well as Excel can communicate with each other and use each other's capabilities. The most common is Dynamic Data Exchange: although strongly deprecated by Microsoft, this is a common method to send data between applications running on Windows, with official MS publications referring to it as "the protocol from hell". As the name suggests, it allows applications to supply data to others for calculation and display. It is very common in financial markets, being used to connect to important financial data services such as Bloomberg and Reuters.

OLE Object Linking and Embedding allows a Windows application to control another to enable it to format or calculate data. This may take on the form of "embedding" where an application uses another to handle a task that it is more suited to, for example a PowerPoint presentation may be embedded in an Excel spreadsheet or vice versa.

Using external data

Excel users can access external data sources via Microsoft Office features such as (for example) .odc connections built with the Office Data Connection file format. Excel files themselves may be updated using a Microsoft supplied ODBC driver.

Excel can accept data in real-time through several programming interfaces, which allow it to communicate with many data sources such as Bloomberg and Reuters (through addins such as Power Plus Pro).

• DDE: "Dynamic Data Exchange" uses the message passing mechanism in Windows to allow data to flow between Excel and other applications. Although it is easy for users to create such links, programming such links reliably is so difficult that Microsoft, the creators of the system, officially refer to it as "the protocol from hell". In spite of its many issues DDE remains the most common way for data to reach traders in financial markets.
• Network DDE Extended the protocol to allow spreadsheets on different computers to exchange data. Starting with Windows Vista, Microsoft no longer supports the facility.
• Real Time Data: RTD although in many ways technically superior to DDE, has been slow to gain acceptance, since it requires non-trivial programming skills, and when first released was neither adequately documented nor supported by the major data vendors.

Alternatively, Microsoft Query provides ODBC-based browsing within Microsoft Excel.

Export and migration of spreadsheets

Programmers have produced APIs to open Excel spreadsheets in a variety of applications and environments other than Microsoft Excel. These include opening Excel documents on the web using either ActiveX controls, or plugins like the Adobe Flash Player. The Apache POI opensource project provides Java libraries for reading and writing Excel spreadsheet files.

Password protection

Main article: Microsoft Office password protection

Microsoft Excel protection offers several types of passwords:

• Password to open a document
• Password to modify a document
• Password to unprotect the worksheet
• Password to protect workbook
• Password to protect the sharing workbook

All passwords except password to open a document can be removed instantly regardless of the Microsoft Excel version used to create the document. These types of passwords are used primarily for shared work on a document. Such password-protected documents are not encrypted, and a data sources from a set password are saved in a document's header. Password to protect workbook is an exception – when it is set, a document is encrypted with the standard password "VelvetSweatshop", but since it is known to the public, it actually does not add any extra protection to the document. The only type of password that can prevent a trespasser from gaining access to a document is the password to open a document. The cryptographic strength of this kind of protection depends strongly on the Microsoft Excel version that was used to create the document.

In Microsoft Excel 95 and earlier versions, the password to open is converted to a 16-bit key that can be instantly cracked. In Excel 97/2000 the password is converted to a 40-bit key, which can also be cracked very quickly using modern equipment. As regards services that use rainbow tables (e.g. Password-Find), it takes up to several seconds to remove protection. In addition, password-cracking programs can brute-force attack passwords at a rate of hundreds of thousands of passwords a second, which not only lets them decrypt a document but also find the original password.

In Excel 2003/XP the encryption is slightly better – a user can choose any encryption algorithm that is available in the system (see Cryptographic Service Provider). Due to the CSP, an Excel file cannot be decrypted, and thus the password to open cannot be removed, though the brute-force attack speed remains quite high. Nevertheless, the older Excel 97/2000 algorithm is set by the default. Therefore, users who do not change the default settings lack reliable protection of their documents.

The situation changed fundamentally in Excel 2007, where the modern AES algorithm with a key of 128 bits started being used for decryption, and a 50,000-fold use of the hash function SHA1 reduced the speed of brute-force attacks down to hundreds of passwords per second. In Excel 2010, the strength of the protection by the default was increased two times due to the use of a 100,000-fold SHA1 to convert a password to a key.

Other platforms

Excel for mobile

Excel Mobile is a spreadsheet program that can edit XLSX files. It can edit and format text in cells, calculate formulas, search within the spreadsheet, sort rows and columns, freeze panes, filter the columns, add comments, and create charts. It cannot add columns or rows except at the edge of the document, rearrange columns or rows, delete rows or columns, or add spreadsheet tabs. The 2007 version has the ability to use a full-screen mode to deal with limited screen resolution, as well as split panes to view different parts of a worksheet at one time. Protection settings, zoom settings, autofilter settings, certain chart formatting, hidden sheets, and other features are not supported on Excel Mobile, and will be modified upon opening and saving a workbook. In 2015, Excel Mobile became available for Windows 10 and Windows 10 Mobile on Windows Store.

Excel for the web

Further information: Microsoft Office § Office on the web

Excel for the web is a free lightweight version of Microsoft Excel available as part of Office on the web, which also includes web versions of Microsoft Word and Microsoft PowerPoint.

Excel for the web can display most of the features available in the desktop versions of Excel, although it may not be able to insert or edit them. Certain data connections are not accessible on Excel for the web, including with charts that may use these external connections. Excel for the web also cannot display legacy features, such as Excel 4.0 macros or Excel 5.0 dialog sheets. There are also small differences between how some of the Excel functions work.

Microsoft Excel Viewer

Microsoft Excel Viewer was a freeware program for Microsoft Windows for viewing and printing spreadsheet documents created by Excel. Microsoft retired the viewer in April 2018 with the last security update released in February 2019 for Excel Viewer 2007 (SP3).

The first version released by Microsoft was Excel 97 Viewer. Excel 97 Viewer was supported in Windows CE for Handheld PCs. In October 2004, Microsoft released Excel Viewer 2003. In September 2007, Microsoft released Excel Viewer 2003 Service Pack 3 (SP3). In January 2008, Microsoft released Excel Viewer 2007 (featuring a non-collapsible Ribbon interface). In April 2009, Microsoft released Excel Viewer 2007 Service Pack 2 (SP2). In October 2011, Microsoft released Excel Viewer 2007 Service Pack 3 (SP3).

Microsoft advises to view and print Excel files for free to use the Excel Mobile application for Windows 10 and for Windows 7 and Windows 8 to upload the file to OneDrive and use Excel for the web with a Microsoft account to open them in a browser.

Quirks

In addition to issues with spreadsheets in general, other problems specific to Excel include numeric precision, misleading statistics functions, mod function errors, date limitations and more.

Numeric precision

Main article: Numeric precision in Microsoft Excel

Excel maintains 15 figures in its numbers, but they are not always accurate: the bottom line should be the same as the top line.

Despite the use of 15-figure precision, Excel can display many more figures (up to thirty) upon user request. But the displayed figures are not those actually used in its computations, and so, for example, the difference of two numbers may differ from the difference of their displayed values. Although such departures are usually beyond the 15th decimal, exceptions do occur, especially for very large or very small numbers. Serious errors can occur if decisions are made based upon automated comparisons of numbers (for example, using the Excel If function), as equality of two numbers can be unpredictable.

In the figure, the fraction 1/9000 is displayed in Excel. Although this number has a decimal representation that is an infinite string of ones, Excel displays only the leading 15 figures. In the second line, the number one is added to the fraction, and again Excel displays only 15 figures. In the third line, one is subtracted from the sum using Excel. Because the sum in the second line has only eleven 1's after the decimal, the difference when 1 is subtracted from this displayed value is three 0's followed by a string of eleven 1's. However, the difference reported by Excel in the third line is three 0's followed by a string of thirteen 1's and two extra erroneous digits. This is because Excel calculates with about half a digit more than it displays.

Excel works with a modified 1985 version of the IEEE 754 specification. Excel's implementation involves conversions between binary and decimal representations, leading to accuracy that is on average better than one would expect from simple fifteen digit precision, but that can be worse. See the main article for details.

Besides accuracy in user computations, the question of accuracy in Excel-provided functions may be raised. Particularly in the arena of statistical functions, Excel has been criticized for sacrificing accuracy for speed of calculation.

As many calculations in Excel are executed using VBA, an additional issue is the accuracy of VBA, which varies with variable type and user-requested precision.

Statistical functions

The accuracy and convenience of statistical tools in Excel has been criticized, as mishandling missing data, as returning incorrect values due to inept handling of round-off and large numbers, as only selectively updating calculations on a spreadsheet when some cell values are changed, and as having a limited set of statistical tools. Microsoft has announced some of these issues are addressed in Excel 2010.

Excel MOD function error

Excel has issues with modulo operations. In the case of excessively large results, Excel will return the error warning #NUM! instead of an answer.

Fictional leap day in the year 1900

Excel includes February 29, 1900, incorrectly treating 1900 as a leap year, even though e.g. 2100 is correctly treated as a non-leap year. The bug originated from Lotus 1-2-3 (deliberately implemented to save computer memory), and was also purposely implemented in Excel, for the purpose of bug compatibility. This legacy has later been carried over into Office Open XML file format.

Thus a (not necessarily whole) number greater than or equal to 61 interpreted as a date and time are the (real) number of days after December 30, 1899, 0:00, a non-negative number less than 60 is the number of days after December 31, 1899, 0:00, and numbers with whole part 60 represent the fictional day.

Date range

Excel supports dates with years in the range 1900–9999, except that December 31, 1899, can be entered as 0 and is displayed as 0-jan-1900.

Converting a fraction of a day into hours, minutes and days by treating it as a moment on the day January 1, 1900, does not work for a negative fraction.

Conversion problems

Entering text that happens to be in a form that is interpreted as a date, the text can be unintentionally changed to a standard date format. A similar problem occurs when a text happens to be in the form of a floating-point notation of a number. In these cases the original exact text cannot be recovered from the result. Formatting the cell as TEXT before entering ambiguous text prevents Excel from converting to a date.

This issue has caused a well known problem in the analysis of DNA, for example in bioinformatics. As first reported in 2004, genetic scientists found that Excel automatically and incorrectly converts certain gene names into dates. A follow-up study in 2016 found many peer reviewed scientific journal papers had been affected and that "Of the selected journals, the proportion of published articles with Excel files containing gene lists that are affected by gene name errors is 19.6 %." Excel parses the copied and pasted data and sometimes changes them depending on what it thinks they are. For example, MARCH1 (Membrane Associated Ring-CH-type finger 1) gets converted to the date March 1 (1-Mar) and SEPT2 (Septin 2) is converted into September 2 (2-Sep) etc. While some secondary news sources reported this as a fault with Excel, the original authors of the 2016 paper placed the blame with the researchers misusing Excel.

In August 2020 the HUGO Gene Nomenclature Committee (HGNC) published new guidelines in the journal Nature regarding gene naming in order to avoid issues with "symbols that affect data handling and retrieval." So far 27 genes have been renamed, including changing MARCH1 to MARCHF1 and SEPT1 to SEPTIN1 in order to avoid accidental conversion of the gene names into dates.

In October 2023, Microsoft fixed the long-standing issue.

Errors with large strings

The following functions return incorrect results when passed a string longer than 255 characters:

• type() incorrectly returns 16, meaning "Error value"
• IsText(), when called as a method of the VBA object WorksheetFunction (i.e., WorksheetFunction.IsText() in VBA), incorrectly returns "false".

Filenames

Microsoft Excel will not open two documents with the same name and instead will display the following error:

A document with the name '%s' is already open. You cannot open two documents with the same name, even if the documents are in different folders. To open the second document, either close the document that is currently open, or rename one of the documents.

The reason is for calculation ambiguity with linked cells. If there is a cell ='[Book1.xlsx]Sheet1'!$G$33, and there are two books named "Book1" open, there is no way to tell which one the user means.

Versions

Early history

Microsoft originally marketed a spreadsheet program called Multiplan in 1982. Multiplan became very popular on CP/M systems, but on MS-DOS systems it lost popularity to Lotus 1-2-3. Microsoft released the first version of Excel for the Macintosh on September 30, 1985, and the first Windows version was 2.05 (to synchronize with the Macintosh version 2.2) on November 19, 1987. Lotus was slow to bring 1-2-3 to Windows and by the early 1990s, Excel had started to outsell 1-2-3 and helped Microsoft achieve its position as a leading PC software developer. This accomplishment solidified Microsoft as a valid competitor and showed its future in developing GUI software. Microsoft maintained its advantage with regular new releases, every two years or so.

Microsoft Windows

Excel 2.0 is the first version of Excel for the Intel platform. Versions prior to 2.0 were only available on the Apple Macintosh.

Excel 2.0 (1987)

The first Windows version was labeled "2" to correspond to the Mac version. It was announced on October 6, 1987, and released on November 19. This included a run-time version of Windows.

BYTE in 1989 listed Excel for Windows as among the "Distinction" winners of the BYTE Awards. The magazine stated that the port of the "extraordinary" Macintosh version "shines", with a user interface as good as or better than the original.

Excel 3.0 (1990)

Included toolbars, drawing capabilities, outlining, add-in support, 3D charts, and many more new features.

Excel 4.0 (1992)

Introduced auto-fill.

Also, an easter egg in Excel 4.0 reveals a hidden animation of a dancing set of numbers 1 through 3, representing Lotus 1-2-3, which is then crushed by an Excel logo.

Excel 5.0 (1993)

With version 5.0, Excel has included Visual Basic for Applications (VBA), a programming language based on Visual Basic which adds the ability to automate tasks in Excel and to provide user-defined functions (UDF) for use in worksheets. VBA includes a fully featured integrated development environment (IDE). Macro recording can produce VBA code replicating user actions, thus allowing simple automation of regular tasks. VBA allows the creation of forms and in‑worksheet controls to communicate with the user. The language supports use (but not creation) of ActiveX (COM) DLL's; later versions add support for class modules allowing the use of basic object-oriented programming techniques.

The automation functionality provided by VBA made Excel a target for macro viruses. This caused serious problems until antivirus products began to detect these viruses. Microsoft belatedly took steps to prevent the misuse by adding the ability to disable macros completely, to enable macros when opening a workbook or to trust all macros signed using a trusted certificate.

Versions 5.0 to 9.0 of Excel contain various Easter eggs, including a "Hall of Tortured Souls", a Doom-like minigame, although since version 10 Microsoft has taken measures to eliminate such undocumented features from their products.

5.0 was released in a 16-bit x86 version for Windows 3.1 and later in a 32-bit version for NT 3.51 (x86/Alpha/PowerPC)

Excel 95 (v7.0)

Microsoft Excel 95

Released in 1995 with Microsoft Office for Windows 95, this is the first major version after Excel 5.0, as there is no Excel 6.0 with all of the Office applications standardizing on the same major version number.

Internal rewrite to 32-bits. Almost no external changes, but faster and more stable.

Excel 95 contained a hidden Doom-like mini-game called "The Hall of Tortured Souls", a series of rooms featuring the names and faces of the developers as an easter egg.

Excel 97 (v8.0)

Included in Office 97 (for x86 and Alpha). This was a major upgrade that introduced the paper clip office assistant and featured standard VBA used instead of internal Excel Basic. It introduced the now-removed Natural Language labels.

This version of Excel includes a flight simulator as an Easter Egg.

Excel 2000 (v9.0)

Microsoft Excel 2000

Included in Office 2000. This was a minor upgrade but introduced an upgrade to the clipboard where it can hold multiple objects at once. The Office Assistant, whose frequent unsolicited appearance in Excel 97 had annoyed many users, became less intrusive.

A small 3-D game called "Dev Hunter" (inspired by Spy Hunter) was included as an easter egg.

Excel 2002 (v10.0)

Included in Office XP. Very minor enhancements.

Excel 2003 (v11.0)

Included in Office 2003. Minor enhancements.

Excel 2007 (v12.0)

Microsoft Excel 2007

Included in Office 2007. This release was a major upgrade from the previous version. Similar to other updated Office products, Excel in 2007 used the new Ribbon menu system. This was different from what users were used to, and was met with mixed reactions. One study reported fairly good acceptance by users except for highly experienced users and users of word processing applications with a classical WIMP interface, but was less convinced in terms of efficiency and organization. However, an online survey reported that a majority of respondents had a negative opinion of the change, with advanced users being "somewhat more negative" than intermediate users, and users reporting a self-estimated reduction in productivity.

Added functionality included Tables, and the SmartArt set of editable business diagrams. Also added was an improved management of named variables through the Name Manager, and much-improved flexibility in formatting graphs, which allow (x, y) coordinate labeling and lines of arbitrary weight. Several improvements to pivot tables were introduced.

Also like other office products, the Office Open XML file formats were introduced, including .xlsm for a workbook with macros and .xlsx for a workbook without macros.

Specifically, many of the size limitations of previous versions were greatly increased. To illustrate, the number of rows was now 1,048,576 (2²⁰) and the columns was 16,384 (2¹⁴; the far-right column is XFD). This changes what is a valid A1 reference versus a named range. This version made more extensive use of multiple cores for the calculation of spreadsheets; however, VBA macros are not handled in parallel and XLL add‑ins were only executed in parallel if they were thread-safe and this was indicated at registration.

Excel 2010 (v14.0)

Microsoft Excel 2010 running on Windows 7

Included in Office 2010, this is the next major version after v12.0, as version number 13 was skipped.

Minor enhancements and 64-bit support, including the following:

• Multi-threading recalculation (MTR) for commonly used functions
• Improved pivot tables
• More conditional formatting options
• Additional image editing capabilities
• In-cell charts called sparklines
• Ability to preview before pasting
• Office 2010 backstage feature for document-related tasks
• Ability to customize the Ribbon
• Many new formulas, most highly specialized to improve accuracy

Excel 2013 (v15.0)

Included in Office 2013, along with a lot of new tools included in this release:

• Improved Multi-threading and Memory Contention
• FlashFill
• Power View
• Power Pivot
• Timeline Slicer
• Windows App
• Inquire
• 50 new functions

Excel 2016 (v16.0)

Included in Office 2016, along with a lot of new tools included in this release:

• Power Query integration
• Read-only mode for Excel
• Keyboard access for Pivot Tables and Slicers in Excel
• New Chart Types
• Quick data linking in Visio
• Excel forecasting functions
• Support for multiselection of Slicer items using touch
• Time grouping and Pivot Chart Drill Down
• Excel data cards

Excel 2019, Excel 2021, Office 365 and subsequent (v16.0)

Microsoft no longer releases Office or Excel in discrete versions. Instead, features are introduced automatically over time using Windows Update. The version number remains 16.0. Thereafter only the approximate dates when features appear can now be given.

• Dynamic Arrays. These are essentially Array Formulas but they "Spill" automatically into neighboring cells and do not need the ctrl-shift-enter to create them. Further, dynamic arrays are the default format, with new "@" and "#" operators to provide compatibility with previous versions. This is perhaps the biggest structural change since 2007, and is in response to a similar feature in Google Sheets. Dynamic arrays started appearing in pre-releases about 2018, and as of March 2020 are available in published versions of Office 365 provided a user selected "Office Insiders".

Apple Macintosh

Microsoft Excel for Mac 2011

• 1985 Excel 1.0
• 1988 Excel 1.5
• 1989 Excel 2.2
• 1990 Excel 3.0
• 1992 Excel 4.0
• 1993 Excel 5.0 (part of Office 4.x—Final Motorola 680x0 version and first PowerPC version)
• 1998 Excel 8.0 (part of Office 98)
• 2000 Excel 9.0 (part of Office 2001)
• 2001 Excel 10.0 (part of Office v. X)
• 2004 Excel 11.0 (part of Office 2004)
• 2008 Excel 12.0 (part of Office 2008)
• 2010 Excel 14.0 (part of Office 2011)
• 2015 Excel 15.0 (part of Office 2016—Office 2016 for Mac brings the Mac version much closer to parity with its Windows cousin, harmonizing many of the reporting and high-level developer functions, while bringing the ribbon and styling into line with its PC counterpart.)

OS/2

• 1989 Excel 2.2
• 1990 Excel 2.3
• 1991 Excel 3.0

Summary

Legend:	Old version, not maintained	Older version, still maintained	Current stable version

Microsoft Excel for Windows release history

Year	Name	Version	Comments
1987	Excel 2	2.0	Renumbered to 2 to correspond with contemporary Macintosh version. Supported macros (later known as Excel 4 macros).
1990	Excel 3	3.0	Added 3D graphing capabilities
1992	Excel 4	4.0	Introduced auto-fill feature
1993	Excel 5	5.0	Included Visual Basic for Applications (VBA) and various object-oriented options
1995	Excel 95	7.0	Renumbered for contemporary Word version. Both programs were packaged in Microsoft Office by this time.
1997	Excel 97	8.0
2000	Excel 2000	9.0	Part of Microsoft Office 2000, which was itself part of Windows Millennium (also known as "Windows ME").
2002	Excel 2002	10.0
2003	Excel 2003	11.0	Released only 1 year later to correspond better with the rest of Microsoft Office (Word, PowerPoint, etc.).
2007	Excel 2007	12.0
2010	Excel 2010	14.0	Due to superstitions surrounding the number 13, Excel 13 was skipped in version counting.
2013	Excel 2013	15.0	Introduced 50 more mathematical functions (available as pre-packaged commands, rather than typing the formula manually).
2016	Excel 2016	16.0	Part of Microsoft Office 2016

Microsoft Excel for Macintosh release history

Year	Name	Version	Comments
1985	Excel 1	1.0	Initial version of Excel. Supported macros (later known as Excel 4 macros).
1988	Excel 1.5	1.5
1989	Excel 2	2.2
1990	Excel 3	3.0
1992	Excel 4	4.0
1993	Excel 5	5.0	Only available on PowerPC-based Macs. First PowerPC version.
1998	Excel 98	8.0	Excel 6 and Excel 7 were skipped to correspond with the rest of Microsoft Office at the time.
2000	Excel 2000	9.0
2001	Excel 2001	10.0
2004	Excel 2004	11.0
2008	Excel 2008	12.0
2011	Excel 2011	14.0	As with the Windows version, version 13 was skipped for superstitious reasons.
2016	Excel 2016	16.0	As with the rest of Microsoft Office, so it is for Excel: Future release dates for the Macintosh version are intended to correspond better to those for the Windows version, from 2016 onward.

Microsoft Excel for OS/2 release history

Year	Name	Version	Comments
1989	Excel 2.2	2.2	Numbered in between Windows versions at the time
1990	Excel 2.3	2.3
1991	Excel 3	3.0	Last OS/2 version. Discontinued subseries of Microsoft Excel, which is otherwise still an actively developed program.

Impact

Excel offers many user interface tweaks over the earliest electronic spreadsheets; however, the essence remains the same as in the original spreadsheet software, VisiCalc: the program displays cells organized in rows and columns, and each cell may contain data or a formula, with relative or absolute references to other cells.

Excel 2.0 for Windows, which was modeled after its Mac GUI-based counterpart, indirectly expanded the installed base of the then-nascent Windows environment. Excel 2.0 was released a month before Windows 2.0, and the installed base of Windows was so low at that point in 1987 that Microsoft had to bundle a runtime version of Windows 1.0 with Excel 2.0. Unlike Microsoft Word, there never was a DOS version of Excel.

Excel became the first spreadsheet to allow the user to define the appearance of spreadsheets (fonts, character attributes, and cell appearance). It also introduced intelligent cell re-computation, where only cells dependent on the cell being modified are updated (previous spreadsheet programs recomputed everything all the time or waited for a specific user command). Excel introduced auto-fill, the ability to drag and expand the selection box to automatically copy a cell or row contents to adjacent cells or rows, adjusting the copies intelligently by automatically incrementing cell references or contents. Excel also introduced extensive graphing capabilities.

Security

Because Excel is widely used, it has been attacked by hackers. While Excel is not directly exposed to the Internet, if an attacker can get a victim to open a file in Excel, and there is an appropriate security bug in Excel, then the attacker can gain control of the victim's computer. UK's GCHQ has a tool named TORNADO ALLEY with this purpose.

Games

Besides the easter eggs, numerous games have been created or recreated in Excel, such as Tetris, 2048, Scrabble, Yahtzee, Angry Birds, Pac-Man, Civilization, Monopoly, Battleship, Blackjack, Space Invaders, and others.

In 2020, Excel became an esport with the advent of the Financial Modeling World Cup.

Numerical Recipes

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Numerical_Recipes 

 

Numerical Recipes: The Art of Scientific Computing

Cover of the third (C++) edition
Author	William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery
Language	English
Discipline	Numerical analysis
Publisher	Cambridge University Press
Website	numerical.recipes

Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1986. The most recent edition was published in 2007.

Overview

The Numerical Recipes books cover a range of topics that include both classical numerical analysis (interpolation, integration, linear algebra, differential equations, and so on), signal processing (Fourier methods, filtering), statistical treatment of data, and a few topics in machine learning (hidden Markov model, support vector machines). The writing style is accessible and has an informal tone. The emphasis is on understanding the underlying basics of techniques, not on the refinements that may, in practice, be needed to achieve optimal performance and reliability. Few results are proved with any degree of rigor, although the ideas behind proofs are often sketched, and references are given. Importantly, virtually all methods that are discussed are also implemented in a programming language, with the code printed in the book. Each version is keyed to a specific language.

According to the publisher, Cambridge University Press, the Numerical Recipes books are historically the all-time best-selling books on scientific programming methods. In recent years, Numerical Recipes books have been cited in the scientific literature more than 3000 times per year according to ISI Web of Knowledge (e.g., 3962 times in the year 2008). And as of the end of 2017, the book had over 44000 citations on Google Scholar.

History

The first publication was in 1986 with the title,”Numerical Recipes, The Art of Scientific Computing”, containing code in both Fortran and Pascal; an accompanying book, “Numerical Recipes Example Book (Pascal)” was first published in 1985. (A preface note in “Examples" mentions that the main book was also published in 1985, but the official note in that book says 1986.) Supplemental editions followed with code in Pascal, BASIC, and C. Numerical Recipes took, from the start, an opinionated editorial position at odds with the conventional wisdom of the numerical analysis community:

If there is a single dominant theme in this book, it is that practical methods of numerical computation can be simultaneously efficient, clever, and — important — clear. The alternative viewpoint, that efficient computational methods must necessarily be so arcane and complex as to be useful only in "black box" form, we firmly reject.

However, as it turned out, the 1980s were fertile years for the "black box" side, yielding important libraries such as BLAS and LAPACK, and integrated environments like MATLAB and Mathematica. By the early 1990s, when Second Edition versions of Numerical Recipes (with code in C, Fortran-77, and Fortran-90) were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between the more mathematical numerical analysts and the larger community using integrated environments. The Second Edition versions occupied a stable role in this niche environment.

By the mid-2000s, the practice of scientific computing had been radically altered by the mature Internet and Web. Recognizing that their Numerical Recipes books were increasingly valued more for their explanatory text than for their code examples, the authors significantly expanded the scope of the book, and significantly rewrote a large part of the text. They continued to include code, still printed in the book, now in C++, for every method discussed. The Third Edition was also released as an electronic book, eventually made available on the Web for free (with nags) or by paid or institutional subscription (with faster, full access and no nags).

In 2015 Numerical Recipes sold its historic two-letter domain name nr.com and became numerical.recipes instead.

Reception

Content

Numerical Recipes is a single volume that covers very broad range of algorithms. Unfortunately that format skewed the choice of algorithms towards simpler and shorter early algorithms which were not as accurate, efficient or stable as later more complex algorithms. The first edition had also some minor bugs, which were fixed in later editions; however according to the authors for years they were encountering on the internet rumors that Numerical Recipes is "full of bugs". They attributed this to people using outdated versions of the code, bugs in other parts of the code and misuse of routines which require some understanding to use correctly.

The rebuttal does not, however, cover criticisms regarding lack of mentions to code limitations, boundary conditions, and more modern algorithms, another theme in Snyder's comment compilation. A precision issue in Bessel functions has persisted to the third edition according to Pavel Holoborodko.

Despite criticism by numerical analysts, engineers and scientists generally find the book conveniently broad in scope. Norman Gray concurs in the following quote:

Numerical Recipes [nr] does not claim to be a numerical analysis textbook, and it makes a point of noting that its authors are (astro-)physicists and engineers rather than analysts, and so share the motivations and impatience of the book's intended audience. The declared premise of the NR authors is that you will come to grief one way or the other if you use numerical routines you do not understand. They attempt to give you enough mathematical detail that you understand the routines they present, in enough depth that you can diagnose problems when they occur, and make more sophisticated choices about replacements when the NR routines run out of steam. Problems will occur because [...]

License

The code listings are copyrighted and commercially licensed by the Numerical Recipes authors. A license to use the code is given with the purchase of a book, but the terms of use are highly restrictive. For example, programmers need to make sure NR code cannot be extracted from their finished programs and used – a difficult requirement with dubious enforceability.

However, Numerical Recipes does include the following statement regarding copyrights on computer programs:

Copyright does not protect ideas, but only the expression of those ideas in a particular form. In the case of a computer program, the ideas consist of the program's methodology and algorithm, including the necessary sequence of steps adopted by the programmer. The expression of those ideas is the program source code ... If you analyze the ideas contained in a program, and then express those ideas in your own completely different implementation, then that new program implementation belongs to you.

One early motivation for the GNU Scientific Library was that a free library was needed as a substitute for Numerical Recipes.

Style

Another line of criticism centers on the coding style of the books, which strike some modern readers as "Fortran-ish", though written in contemporary, object-oriented C++. The authors have defended their very terse coding style as necessary to the format of the book because of space limitations and for readability.

Titles in the series (partial list)

The books differ by edition (1st, 2nd, and 3rd) and by the computer language in which the code is given.

• Numerical Recipes. The Art of Scientific Computing, 1st Edition, 1986, ISBN 0-521-30811-9. (Fortran and Pascal)
• Numerical Recipes in C. The Art of Scientific Computing, 1st Edition, 1988, ISBN 0-521-35465-X.
• Numerical Recipes in Pascal. The Art of Scientific Computing, 1st Edition, 1989, ISBN 0-521-37516-9.
• Numerical Recipes in Fortran. The Art of Scientific Computing, 1st Edition, 1989, ISBN 0-521-38330-7.
• Numerical Recipes in BASIC. The Art of Scientific Computing, 1st Edition, 1991, ISBN 0-521-40689-7. (supplemental edition)
• Numerical Recipes in Fortran 77. The Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43064-X.
• Numerical Recipes in C. The Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43108-5.
• Numerical Recipes in Fortran 90. The Art of Parallel Scientific Computing, 2nd Edition, 1996, ISBN 0-521-57439-0.
• Numerical Recipes in C++. The Art of Scientific Computing, 2nd Edition, 2002, ISBN 0-521-75033-4.
• Numerical Recipes. The Art of Scientific Computing, 3rd Edition, 2007, ISBN 0-521-88068-8. (C++ code)

The books are published by Cambridge University Press.

Numerical analysis

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Numerical_analysis

Babylonian clay tablet YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/60² + 10/60³ = 1.41421296...

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.

Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas continue to be used in software algorithms.

The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.

Numerical analysis continues this long tradition: rather than giving exact symbolic answers translated into digits and applicable only to real-world measurements, approximate solutions within specified error bounds are used.

General introduction

The overall goal of the field of numerical analysis is the design and analysis of techniques to give approximate but accurate solutions to hard problems, the variety of which is suggested by the following:

• Advanced numerical methods are essential in making numerical weather prediction feasible.
• Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of ordinary differential equations.
• Car companies can improve the crash safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving partial differential equations numerically.
• Hedge funds (private investment funds) use quantitative finance tools from numerical analysis to attempt to calculate the value of stocks and derivatives more precisely than other market participants.
• Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs. Historically, such algorithms were developed within the overlapping field of operations research.
• Insurance companies use numerical programs for actuarial analysis.

The rest of this section outlines several important themes of numerical analysis.

History

The field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method. The origins of modern numerical analysis are often linked to a 1947 paper by John von Neumann and Herman Goldstine, but others consider modern numerical analysis to go back to work by E. T. Whittaker in 1912.

To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions. The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points. The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy.

The mechanical calculator was also developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

The Leslie Fox Prize for Numerical Analysis was initiated in 1985 by the Institute of Mathematics and its Applications.

Direct and iterative methods

Consider the problem of solving

3x³ + 4 = 28

for the unknown quantity x.

Direct method

	3x3 + 4 = 28.
Subtract 4	3x3 = 24.
Divide by 3	x3 =  8.
Take cube roots	x =  2.

For the iterative method, apply the bisection method to f(x) = 3x³ − 24. The initial values are a = 0, b = 3, f(a) = −24, f(b) = 57.

Iterative method

a	b	mid	f(mid)
0	3	1.5	−13.875
1.5	3	2.25	10.17...
1.5	2.25	1.875	−4.22...
1.875	2.25	2.0625	2.32...

From this table it can be concluded that the solution is between 1.875 and 2.0625. The algorithm might return any number in that range with an error less than 0.2.

Discretization and numerical integration

In a two-hour race, the speed of the car is measured at three instants and recorded in the following table.

Time	0:20	1:00	1:40
km/h	140	150	180

A discretization would be to say that the speed of the car was constant from 0:00 to 0:40, then from 0:40 to 1:20 and finally from 1:20 to 2:00. For instance, the total distance traveled in the first 40 minutes is approximately (2/3 h × 140 km/h) = 93.3 km. This would allow us to estimate the total distance traveled as 93.3 km + 100 km + 120 km = 313.3 km, which is an example of numerical integration (see below) using a Riemann sum, because displacement is the integral of velocity.

Ill-conditioned problem: Take the function f(x) = 1/(x − 1). Note that f(1.1) = 10 and f(1.001) = 1000: a change in x of less than 0.1 turns into a change in f(x) of nearly 1000. Evaluating f(x) near x = 1 is an ill-conditioned problem.

Well-conditioned problem: By contrast, evaluating the same function f(x) = 1/(x − 1) near x = 10 is a well-conditioned problem. For instance, f(10) = 1/9 ≈ 0.111 and f(11) = 0.1: a modest change in x leads to a modest change in f(x).

Direct methods compute the solution to a problem in a finite number of steps. These methods would give the precise answer if they were performed in infinite precision arithmetic. Examples include Gaussian elimination, the QR factorization method for solving systems of linear equations, and the simplex method of linear programming. In practice, finite precision is used and the result is an approximation of the true solution (assuming stability).

In contrast to direct methods, iterative methods are not expected to terminate in a finite number of steps. Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit. A convergence test, often involving the residual, is specified in order to decide when a sufficiently accurate solution has (hopefully) been found. Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general). Examples include Newton's method, the bisection method, and Jacobi iteration. In computational matrix algebra, iterative methods are generally needed for large problems.

Iterative methods are more common than direct methods in numerical analysis. Some methods are direct in principle but are usually used as though they were not, e.g. GMRES and the conjugate gradient method. For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method.

Discretization

Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called 'discretization'. For example, the solution of a differential equation is a function. This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

Generation and propagation of errors

The study of errors forms an important part of numerical analysis. There are several ways in which error can be introduced in the solution of the problem.

Round-off

Round-off errors arise because it is impossible to represent all real numbers exactly on a machine with finite memory (which is what all practical digital computers are).

Truncation and discretization error

Truncation errors are committed when an iterative method is terminated or a mathematical procedure is approximated and the approximate solution differs from the exact solution. Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem. In the example above to compute the solution of ${\displaystyle 3x^3+4=28}$, after ten iterations, the calculated root is roughly 1.99. Therefore, the truncation error is roughly 0.01.

Once an error is generated, it propagates through the calculation. For example, the operation + on a computer is inexact. A calculation of the type ${\displaystyle a+b+c+d+e}$ is even more inexact.

A truncation error is created when a mathematical procedure is approximated. To integrate a function exactly, an infinite sum of regions must be found, but numerically only a finite sum of regions can be found, and hence the approximation of the exact solution. Similarly, to differentiate a function, the differential element approaches zero, but numerically only a nonzero value of the differential element can be chosen.

Numerical stability and well-posed problems

Numerical stability is a notion in numerical analysis. An algorithm is called 'numerically stable' if an error, whatever its cause, does not grow to be much larger during the calculation. This happens if the problem is 'well-conditioned', meaning that the solution changes by only a small amount if the problem data are changed by a small amount. To the contrary, if a problem is 'ill-conditioned', then any small error in the data will grow to be a large error.

Both the original problem and the algorithm used to solve that problem can be 'well-conditioned' or 'ill-conditioned', and any combination is possible.

So an algorithm that solves a well-conditioned problem may be either numerically stable or numerically unstable. An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem. For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem. Many algorithms solve this problem by starting with an initial approximation x₀ to ${\displaystyle \sqrt{2}}$, for instance x₀ = 1.4, and then computing improved guesses x₁, x₂, etc. One such method is the famous Babylonian method, which is given by x_k+1 = x_k/2 + 1/x_k. Another method, called 'method X', is given by x_k+1 = (x_k² − 2)² + x_k. A few iterations of each scheme are calculated in table form below, with initial guesses x₀ = 1.4 and x₀ = 1.42.

Babylonian	Babylonian	Method X	Method X
x0 = 1.4	x0 = 1.42	x0 = 1.4	x0 = 1.42
x1 = 1.4142857...	x1 = 1.41422535...	x1 = 1.4016	x1 = 1.42026896
x2 = 1.414213564...	x2 = 1.41421356242...	x2 = 1.4028614...	x2 = 1.42056...
		...	...
		x1000000 = 1.41421...	x27 = 7280.2284...

Observe that the Babylonian method converges quickly regardless of the initial guess, whereas Method X converges extremely slowly with initial guess x₀ = 1.4 and diverges for initial guess x₀ = 1.42. Hence, the Babylonian method is numerically stable, while Method X is numerically unstable.

Numerical stability is affected by the number of the significant digits the machine keeps. If a machine is used that keeps only the four most significant decimal digits, a good example on loss of significance can be given by the two equivalent functions

${\displaystyle f(x)=x\left(\sqrt{x+1}-\sqrt{x}\right)}$ and ${\displaystyle g(x)=\frac{x}{\sqrt{x+1}+\sqrt{x}}.}$

Comparing the results of

${\displaystyle f(500)=500\left(\sqrt{501}-\sqrt{500}\right)=500\left(22.38-22.36\right)=500(0.02)=10}$

and

${\displaystyle \begin{array}{rl}g(500)& =\frac{500}{\sqrt{501}+\sqrt{500}}\\ & =\frac{500}{22.38+22.36}\\ & =\frac{500}{44.74}=11.17\end{array}}$

by comparing the two results above, it is clear that loss of significance (caused here by catastrophic cancellation from subtracting approximations to the nearby numbers ${\displaystyle \sqrt{501}}$ and ${\displaystyle \sqrt{500}}$, despite the subtraction being computed exactly) has a huge effect on the results, even though both functions are equivalent, as shown below

${\displaystyle \begin{array}{rl}f(x)& =x\left(\sqrt{x+1}-\sqrt{x}\right)\\ & =x\left(\sqrt{x+1}-\sqrt{x}\right)\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}\\ & =x\frac{(\sqrt{x+1}{)}^2-(\sqrt{x}{)}^2}{\sqrt{x+1}+\sqrt{x}}\\ & =x\frac{x+1-x}{\sqrt{x+1}+\sqrt{x}}\\ & =x\frac{1}{\sqrt{x+1}+\sqrt{x}}\\ & =\frac{x}{\sqrt{x+1}+\sqrt{x}}\\ & =g(x)\end{array}}$

The desired value, computed using infinite precision, is 11.174755...

• The example is a modification of one taken from Mathew; Numerical methods using MATLAB, 3rd ed.

Areas of study

The field of numerical analysis includes many sub-disciplines. Some of the major ones are:

Computing values of functions

Interpolation: Observing that the temperature varies from 20 degrees Celsius at 1:00 to 14 degrees at 3:00, a linear interpolation of this data would conclude that it was 17 degrees at 2:00 and 18.5 degrees at 1:30pm. Extrapolation: If the gross domestic product of a country has been growing an average of 5% per year and was 100 billion last year, it might extrapolated that it will be 105 billion this year. Regression: In linear regression, given n points, a line is computed that passes as close as possible to those n points. Optimization: Suppose lemonade is sold at a lemonade stand, at $1.00 per glass, that 197 glasses of lemonade can be sold per day, and that for each increase of $0.01, one less glass of lemonade will be sold per day. If $1.485 could be charged, profit would be maximized, but due to the constraint of having to charge a whole-cent amount, charging $1.48 or $1.49 per glass will both yield the maximum income of $220.52 per day. Differential equation: If 100 fans are set up to blow air from one end of the room to the other and then a feather is dropped into the wind, what happens? The feather will follow the air currents, which may be very complex. One approximation is to measure the speed at which the air is blowing near the feather every second, and advance the simulated feather as if it were moving in a straight line at that same speed for one second, before measuring the wind speed again. This is called the Euler method for solving an ordinary differential equation.

One of the simplest problems is the evaluation of a function at a given point. The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient. For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions. Generally, it is important to estimate and control round-off errors arising from the use of floating-point arithmetic.

Interpolation, extrapolation, and regression

Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points?

Extrapolation is very similar to interpolation, except that now the value of the unknown function at a point which is outside the given points must be found.

Regression is also similar, but it takes into account that the data are imprecise. Given some points, and a measurement of the value of some function at these points (with an error), the unknown function can be found. The least squares-method is one way to achieve this.

Solving equations and systems of equations

Another fundamental problem is computing the solution of some given equation. Two cases are commonly distinguished, depending on whether the equation is linear or not. For instance, the equation ${\displaystyle 2x+5=3}$ is linear while ${\displaystyle 2x^2+5=3}$ is not.

Much effort has been put in the development of methods for solving systems of linear equations. Standard direct methods, i.e., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for symmetric (or hermitian) and positive-definite matrix, and QR decomposition for non-square matrices. Iterative methods such as the Jacobi method, Gauss–Seidel method, successive over-relaxation and conjugate gradient method are usually preferred for large systems. General iterative methods can be developed using a matrix splitting.

Root-finding algorithms are used to solve nonlinear equations (they are so named since a root of a function is an argument for which the function yields zero). If the function is differentiable and the derivative is known, then Newton's method is a popular choice. Linearization is another technique for solving nonlinear equations.

Solving eigenvalue or singular value problems

Several important problems can be phrased in terms of eigenvalue decompositions or singular value decompositions. For instance, the spectral image compression algorithm is based on the singular value decomposition. The corresponding tool in statistics is called principal component analysis.

Optimization

Main article: Mathematical optimization

Optimization problems ask for the point at which a given function is maximized (or minimized). Often, the point also has to satisfy some constraints.

The field of optimization is further split in several subfields, depending on the form of the objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A famous method in linear programming is the simplex method.

The method of Lagrange multipliers can be used to reduce optimization problems with constraints to unconstrained optimization problems.

Evaluating integrals

Main article: Numerical integration

Numerical integration, in some instances also known as numerical quadrature, asks for the value of a definite integral. Popular methods use one of the Newton–Cotes formulas (like the midpoint rule or Simpson's rule) or Gaussian quadrature. These methods rely on a "divide and conquer" strategy, whereby an integral on a relatively large set is broken down into integrals on smaller sets. In higher dimensions, where these methods become prohibitively expensive in terms of computational effort, one may use Monte Carlo or quasi-Monte Carlo methods (see Monte Carlo integration), or, in modestly large dimensions, the method of sparse grids.

Differential equations

Main articles: Numerical ordinary differential equations and Numerical partial differential equations

Numerical analysis is also concerned with computing (in an approximate way) the solution of differential equations, both ordinary differential equations and partial differential equations.

Partial differential equations are solved by first discretizing the equation, bringing it into a finite-dimensional subspace. This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method. The theoretical justification of these methods often involves theorems from functional analysis. This reduces the problem to the solution of an algebraic equation.

Software

Main articles: List of numerical-analysis software and Comparison of numerical-analysis software

Since the late twentieth century, most algorithms are implemented in a variety of programming languages. The Netlib repository contains various collections of software routines for numerical problems, mostly in Fortran and C. Commercial products implementing many different numerical algorithms include the IMSL and NAG libraries; a free-software alternative is the GNU Scientific Library.

Over the years the Royal Statistical Society published numerous algorithms in its Applied Statistics (code for these "AS" functions is here); ACM similarly, in its Transactions on Mathematical Software ("TOMS" code is here). The Naval Surface Warfare Center several times published its Library of Mathematics Subroutines (code here).

There are several popular numerical computing applications such as MATLAB, TK Solver, S-PLUS, and IDL as well as free and open source alternatives such as FreeMat, Scilab, GNU Octave (similar to Matlab), and IT++ (a C++ library). There are also programming languages such as R (similar to S-PLUS), Julia, and Python with libraries such as NumPy, SciPy and SymPy. Performance varies widely: while vector and matrix operations are usually fast, scalar loops may vary in speed by more than an order of magnitude.

Many computer algebra systems such as Mathematica also benefit from the availability of arbitrary-precision arithmetic which can provide more accurate results.

Also, any spreadsheet software can be used to solve simple problems relating to numerical analysis. Excel, for example, has hundreds of available functions, including for matrices, which may be used in conjunction with its built in "solver".