beta: w
Answers the value of the beta function defined as
beta(z,w) = gamma(z)gamma(w) / gamma(z + w)
where z is self.
Vector (array of numbers)
Matrices
Matrices2Dimensional
(groups all the two-dimension
matrices)
MatrixND
(groups matrices with three
or more dimensions)
There are a number of additional classes which are special types of
matrices.
Each of the new class adds the basic operations such as addition(+), subtraction(-), and multiplication (*). Within the new class a number of basic math functions have been. Vector respond to mean, mediam, stardardDeviation, variances as well as a number of other useful functions. Matrix responds to inverse, determinant, and many other operations. Complex numbers know how to perform their basic math operations and a few common math functions. These new classes were integrated that mixed mode arithmetic with the the classes is valid. NOTE: Matrices do not support "division".
NOTE: It is recommended that you review the comments in the source code
of the examples and methods (both class and instance) for
additional information. You can also scan the Glossary located at the
end of this manual for an explanation of additional terms used.
Examples
To demonstrate the use of a class and its methods, we have placed examples
in the class methods of the new classes. Each example has comments explaining
its purpose. The example methods are named exampleNN where NN is a two
digit number from 01 to NN. These examples are designed to be evaluated
in a workspace with 'print it.' For instance, to see the results of example03
in the class Matrix, evaluate the following line in a workspace with 'print
it:'
Matrix example03
NOTE: If the class browser does not list example0l, 02, 03, etc., the
method named exampleReadMe will give the name of the class where examples
are to be found. In addition, a number of 'test' methods have been added
to the class methods of existing classes. The methods are prefixed with
'test' and do what the prefix says, "test the method." For example, added
to class Float is testBeta which tests the beta function. Comments are
included in each method explaining its purpose. As with examples, these
test methods are designed to be evaluated in a workspace with 'print it.'
ComplexNumbers, a subclass of Magnitude, groups together
the algebraic form (ComplexNumber) and polar form
(ComplexPolarForm). This choice was made because complex numbers
properties differ significantly from real numbers (members of class Number).
For instance, the inequality operations (< > etc.) are not defined for
complex numbers. The ideal hierarchy would be:
Numbers
RealNumbers (current class Number)
ComplexNumbers
Class ComplexNumber uses the algebraic definition: (a + bi) where a is the real part, b is the imaginary part, and i is the square root of -1.
Class ComplexPolarForm uses the polar form definition: r(cos(theta)
+ i sin(theta)) where r is the modulus (positive number), theta is the
amplitude (angle in radians), and i is the square root of -1.
In this implementation, you should use instances of ComplexNumber
for arithmetic operations (* + - / abs In exp. . .).
ComplexPolarForm is used primarily to find complex roots using
De Moivre's theorem. The few arithmetic operations provided by ComplexPolarForm
change the receiver to a ComplexNumber, perform the operation, and
answer a ComplexNumber.
You can create of a ComplexNumber by sending an instance of Number the message imaginary:. For example, realPart imaginary: imaginaryPart where both realPart and imaginaryPart are Numbers. The shorter message i: has also been provided. You can use themessage cis: to create a complex number in polar form, ComplexPolarForm. For example, modulus cis: amplitude. You can make conversions between ComplexNumber and ComplexPolarForm by using the respective messages asComplexPolarForm and asComplexNumber.
When you use the arithmetic operators, * + - /, the result will always be an instance of ComplexNumber when one of the arguments is a complex number. Appropriate changes have been made to existing Number classes to support complex number arithmetic.
NOTE: These changes do not support operations with an instance of ComplexPolarForm.
ComplexNumber Method Highlights
The following list summarizes most of the common operations used in
ComplexNumbers arithmetic operators, * + - /
amplitude
Answers the angle in radians for the trigonometric
form of the receiver.
asComplexPolarForm
Answers the polar form of the receiver.
complexRoots: anInteger
Answers a Vector with the n, nth roots of the receiver.
conjugate
Answers the conjugate of the receiver.
exp
Answers the exponential of the receiver.
imaginary
Answers the imaginary part of the receiver.
In
Answers the natural log of the receiver.
modulus
Answers the modulus (absolute value) of the receiver.
raisedTo: aNumber
Answers the result of the receiver raised to aNumber
power.
real
Answers the real part of the receiver.
A special version of complexRoots: is shown below as a class method of ComplexNumber
complexRootsOfUnity: anInteger
Answers a Vector containing the n, nth roots of
unity (1).
(Special version of complexRoots:)
Vector, a subclass of Array, implements a special form of an array.
NOTE: While Arrays can contain many different types of objects, Vectors should contain ONLY numbers. Keep in mind, however, that Vector does not force you to use only numbers. Therefore, you must yourself ensure that the Vector subclass contains only numbers.
Another special property of Vectors is that the UndefinedObject nil is allowed in the vector for almost all methods. nil represents missing and/or invalid data values which are not to be used in computations. Missing/invalid values do occur in real-world statistical applications, especially in areas other than the social sciences.
The following is an example using nil. A Vector contains soil
temperature measurements taken from various locations. A few
temperature values appear to be very high compared to the others. The
high values could have been measurement errors or some other abnormality.
Therefore, these high values may have biased the statistics. These values
can be set to nil and the statistics recomputed to determine the amount
of bias introduced by the high values. This technique maintains the original
relationship of the measurements.
For example, if five different measurements were taken at each of 100 sites, the size of the temperature vector stays at 100 when one value is set to nil, meaning that the other four measurements for that location are still at the same relative index location. This results in significant reduction in bookkeeping.
NOTE: Arithmetic operations with a scalar are right-handed operations.
Valid
Wrong
vector * scalar
scalar * vector
Using only right-handed operations requires no changes to the class Number or its subclasses, which results in no additional overhead for arithmetic operations (* + - /).
Vector Method Highlights
Arithmetic messages (* + - /) perform element-wise operations on the
non-nil elements. The messages work with a Vector of the same size
or a scalar.
The following list summarizes of the most common operations used in Vectors.
count
Answers the number of non-nil elements.
minimum
Answers the minimum value of the non-nil elements.
maximum
Answers the maximum value of the non-nil elements.
mean
Answers the mean (average) value of the non-nil
elements.
median
Answers the median value of the non-nil elements.
skewness
Answers the skewness of non-nil values.
standard Deviation
Answers the standard deviation of non-nil values.
variance
Answers the variance of non-nil values.
kurtosis
Answers the kurtosis of non-nil values.
moments
Answers a dictionary containing: count, minimum,
maximum, mean, median, variance, standardDeviation,
absoluteDeviation, skewness, kurtosis.
Messages to Vector where Vector MUST NOT contain any nil elements are as follows:
vector * anObject
Where anObject is a matrix (two-dimensional).
crossProduct: aVector
dotProduct: aVector
Matrices, the root class for all matrices, are subclassed by the number of dimensions into Matrices2Dimensional and MatrixND. The subclasses are as follows:
Matrices
Matrices2Dimensional
Groups the various forms of two-dimensional matrices.
The subclasses are:
LowerTriangularMatrix
Matrix
SquareMatrix
LuMatrix
TriDiagonalMatrix
MatrixND
For matrices with three
or more dimensions. Has no subclasses.
Arithmetic operations expect the matrix to contain ONLY scalars. Otherwise, a matrix may contain any type of object.
NOTE: Matrix operations (* + - /) with a scalar are righthanded operations.
Valid
Wrong
vector * scalar
scalar * vector
Using only right-handed operations requires no changes to the class Number or its subclasses, which results in no additional overhead for arithmetic operations (* + - /).
You can access any form of Matrix through the methods defined
for the specific type of matrix, Matrices2Dimensional or MatrixND
(see the following sections). Methods at: and at:put: access
an element at a linear position regardless of the dimensions of the matrix.
Examples:
• To create a matrix with 12 rows and 4 columns:
Matrix new: 12 @ 4
• To create a matrix with 12 rows and 12 columns:
Matrix new: 12 @ 12
NOTE: Because the number of rows equals the number of columns, a SquareMatrix
is automatically created. This means that in order to create a new instance
of a square or M @ N two-dimensional matrix, you always send the new:
or new:contents: message to Class Matrix.
• To create a matrix with 3 rows and 3 columns and fill the contents
with the given collection:
Matrix new: 3 @ 3 contents: #(1 2 3 4 5 6 7 8 9)
The basic access methods for two-dimensional matrices are as follows:
row: rowNumber col: columnNumber
Answers the element at the given row and column
number.
row: rowNumber col: columnNumber put: anObject
Places anObject at the given row and column number.
columnAt: columnNumber
Answers a Vector containing the elements of the
given columnNumber.
Vector size equals the number of rows for the matrix.
columnAt: columnNumber put: aCollection
Places the collection in the column specified by
columnNumber.
aCollection size must equal the number of rows for
the matrix.
rowAt: rowNumber
Answers a Vector containing the elements of the
given rowNumber.
Vector size equals the number of columns for the
matrix.
rowAt: rowNumber put: aCollection
Places the collection in the row specified by rowNumber.
aCollection size must equal the number of columns
for the matrix.
Two-Dimensional Matrices and Sub-Matrices
A number of manipulation methods exist for changing the structure of
a matrix. Rows or columns may be inserted or removed. Sub-matrices may
be extracted or replaced. A matrix may be appended at the right or bottom
of an existing matrix. These methods are located in Matrices2Dimensional.
They are as follows:
appendAtBottom: anArrayOrMatrix
appendAtRight: anArrayOrMatrix
deleteColumn: anIntegerOrArray
deleteRow: anIntegerOrArray
insertColumns: numberOfColumns before: columnNumber
insertRows: numberOfRows above: rowNumber
row: rowNumber col: columnNumber extractSubMatrix: dimensions
row: rowNumber col: columnNumber placeSubMatrix: aMatrix
See method comments for more explanation.
Matrix Method Highlights
The following list summarizes most of the common operations used in
Matrices.
For an M @ N Matrix (M can equal N)
* aScalarVectorOrMatrix
+ aScalarOrMatrix
- aScalarOrMatrix
= aMatrix
Answers true if the dimensions and contents are
equal.
column: columnNumber1 exchange: columnNumber2
Exchanges contents of columnNumber1 with columnNumber2.
row: rowNumber1 exchange: rowNumber2
Exchanges contents of rowNumber1 with rowNumber2.
transpose
Answers the transpose of the receiver.
For an N x N Matrix (class SquareMatrix)
determinant
inverse
jacobiEigenSystem
Computes both the eigenvalues and eigenvectors of
a
real symmetric matrix by Jacobi transformations.
IuDecomposition
Answers an LuMatrix containing the luDecomposition
of the receiver.
realSymmetricEigenSystem: option
Computes by QL method the eigenvalues and eigenvectors
of a real symmetric matrix depending on the option
chosen.
If the option is nil,
both values and vectors are computed.
If the option is non-nil,
only eigenvalues are computed.
In LuMatrix
determinant
inverse
solveForMatrix: bMatrix
Answers the solution matrix X for A * X = B
where A is the receiver and B is the known matrix.
solveForVector: aVector
Answers the solution vector x for A * x = b
where A is the receiver and b is the known vector.
In TriDiagonalMatrix
computeReaISymmetricEigenSystem: option
Computes by QL method the eigenvalues and eigenvectors
of a real symmetric matrix depending on the option
chosen.
When the option is nil, both values and vectors
are computed.
When the option is non-nil, only eigenvalues are
computed.
Various class example methods are available that demenstrate the matrix
usuage..
MatrixND supports matrices with 3 or more dimensions. The creation
methods are new: aDimensionVector or new:
aDimensionVector contents: aCollection. aDimensionVector can
be an instance of Array or Vector. The total number of dimensions
is aDimensionVector size. The contents of aDimensionVector are integer
values representing the size of each dimension.
Examples:
• To create a matrix with four dimensions where the size of dimensions
one through four are 6, 4,10, 9, respectively:
MatrixND new: #(6 4 10 9)
• To create a three-dimensional matrix with the contents specified by
a Collection:
MatrixND new: #(2 2 2) contents: #(1 2 3 4 5 6 7
8)
NOTE: The collection size must equal the product of the dimension sizes.
In the preceding example, the collection size is 8, which equals 2*2*2.
The primary access methods for MatrixND are as follows:
elementAt: aVector
Answers the element at the position given by the
indices in aVector.
aVector size must equal the number of dimensions
for the MatrixND.
elementAt: aVector put: anObject
Replaces the element at a position given by the
indices in aVector with anObject.
rowAt: indicesVector
Answers a vector containing the row. The desired
row orientation
is designated by the single zero position in the
indicesVector.
rowAt: indicesVector put: aVector
Places the contents of aVector in the desired row.
The desired row orientation is designated by the
single
zero position in the indicesVector.
matrixAt: indicesVector
Answers a matrix extracted from self. The desired
matrix orientation
is designated by two zero positions in the indicesVector.
matrixAt: indicesVector put: aMatrix
Places aMatrix in self. The desired matrix orientation
is designated
by two zero positions in the indicesVector. aMatrix
is assumed to
be two-dimensional.
MatrixND Method Highlights
The following list summarizes most of the common operations used in MatrixND.
* aScalar
+ aScalarOrMatrix
- aScalarOrMatrix
= aMatrix
Answers true if the dimensions and contents are
equal.
The math notation of a function is given in the comments of the functions.
Functions Included
Float additions:
beta: w
Answers the value of the beta function defined as
beta(z,w) = gamma(z)gamma(w) / gamma(z + w)
where z is self.
betalncomplete: aValue b: bValue
Answers the value of the incomplete beta function
defined Ix(a,b), incomplete beta at x for values
(a,b) where x is self.
errorFunction
Answers the special case of the gammaIncomplete
function
errorFunction(x) = gammaIncomplete(0.5,x*x) where
x is self.
errorFunctionComplement
Answers the complementary error function associated
with self errorFunctionComplement(x) = 1 - errorFunction(x)
where x is self.
fRatioSignificance: numberDegreesOfFreedom1 f2: numberDegreesOfFreedom2
Answers the significance of the f-ratio (self) with
the two associated
number of degrees of freedom. The f-ratio is the
larger variance
divided by the smaller variance of the two data
vectors.
numberDegreesOfFreedoml is associated with the larger
variance
and numberDegreesOfFreedom2 with the smaller variance.
Degrees
of freedom are the number of elements (nonnil) in
the respective data vector less one.
Answers with small values to indicate significantly
different variances.
(1.0 - answer) is the F-Distribution Probability
Function value.
(Used for computing F-Test for Significantly Different
Variances.)
gammalncomplete: x
Answers the value for gammaIncomplete(a,x) function
where a is self.
gammalncompleteComplement: x
Answers the value of gammaIncompleteComplement(a,x)function
defined as gammaIncompleteComplement(a,x) = 1 -
gammaIncomplete(a,x)
where a is self.
gammaLn
Answers the In of the gamma function on self.
Integer additions:
binomialCoefficient: k
Answers the binomial coefficient | p | where 0 <=
k <= n and n is self. k! (n-k)! where O>=k<=n
gammaLn
Answers the In of the gamma function for an Integer
value, i.e. n factorial = gamma(n + 1)
Number routes the following messages to Float
beta: betalncomplete: errorFunction
errorFunctionComplement
gammaLn
gammalncomplete
gammalncompleteComplement
Methods were added to Number, ComplexNumber, Vector,
and Matrices to allow formatted printing of numbers. The common
message is #print0n: aStream width: aNumber decimal: numberOfPlaces.
This message requests the receiver to print its contents in ASCII form
on a stream at the current position. Each number printed will be right-justified
in the specified width with the appropriate number of decimal places. Should
the resulting number not fit, asterisks (***) will be printed.
Both Number and ComplexNumber support the additional
message #printWidth:decimal. The receiver answers the ASCII string
of the appropriate width with the specified number of decimal places.
NOTE: When the receiver is an instance of ComplexNumber, the length of the string is two times width plus two. The extra two is for 'i:'.
Similar messages have been provided for instances of String. They are as follows:
print0n: aStream width: anlnteger and
printWidth: anlnteger
Left-justifies the string
and pads the string with blanks on the right.
print0n: aStream rightWidth: anlnteger and
printRightWidth: anlnteger
Right-justifies the string
and pads on the left with blanks.
printWidth: anlnteger and
printRightWidth: anlnteger
Answers a string of width anInteger with the left-
and right-justification, respectively.
String asFloat
Method #asFloat converts a string containing the character representation
of a number to an instance of class Float. The number may be in decimal
notation or a notation. Valid strings are: '1', '1.2', '.2', '2e4', '1E-10'.
Example:
| vectorOfFIoatValues |
vectorOfFIoatValues := '1 .0 6.0 .1 e6' subStrings asVector asFloat.
The string is converted to a vector of float values. When lines in an
ASCII file are numbers (blank delimited), this technique is useful.
Collection
asVector
asMatrix
asMatrix:
SequenceableCollection
doIndex:
findFirst:if None:
Number
isScalar
forceToZero:
printOn:width:decirnal:
printWidth:decimal:
Support for ComplexNumber
asComplexNumber
asComplexPolarForm
cis:
complexRoots:
conjugate
i:
imaginary
imaginary:
pythagorean:
real
Support for Matrices
pseudoAbs
signOf:
Object
isScalar
String
asFloat
printOn:rightWidth:
printOn:width:
printRightWidth:
printWidth:
In cases where there are similarities, the mathematical derivations (formulas) dictate a similar algorithm. The conversion from radian to degrees or degrees to radians requires mathematically identical algorithms. This implies that any implementation will by nature strongly resemble any other implementation.
There are any number of references available for the functionality included
in MathPack. Therefor, only primary references used by the author
are listed.
"Numerical Recipes" is strongly recommended as the best single source
reference. Many of the books give an extensive list of additional references.
These are not repeated here even though many of them were examined. The
references are not listed in any particular order.
Numerical Recipes: The Art of Scientific Computing, William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Cambridge University Press, 1988.
Collected Algorithms From ACM, multiple volumes. Association for Computing Machinery, Inc., 11 West 42nd Street, New York, New York, 10036.
Algorithms, second edition, Robert Sedgewick. Addision Wesley, 1988.
Fundamental Algorithms: The Art of Computer Programming, volume 1, second edition, Donald E. Knuth. Addison-Wesley, 1969.
Seminumerical Algorithms: The Art of Computer Programming, volume 2, second edition, Donald E. Knuth. Addison-Wesley, 1981.
CRC: Standard Mathematical Tables, Twenty-fourth Edition, William H. Beyer. CRC Press, 1976.
Linpack User's Guide, J.J. Dongarra, C.B. Moler, J.R. Bunch, G.W. Stewart. SIAM, 1979. (SIAM, 14th Floor, 117 South 17th Street, Philadelphia, PA 19103-5052.)
Lecture Notes in Computer Science; 6, Matrix Eigensystem Routines
- EISPACK Guide, second edition, B.T. Smith et al,
Springer-Verlag 1976.
Lecture Notes in Computer Science; 51, Matrix Eigensystem Routines
- EISPACK Guide Extension, B.S. Garbow et al.
Springer-Verlag, 1977.
Computational Methods in Elementary Numerical Analysis, John L. Morris. John Wiley & Sons, 1983.
Introduction to Matrices and Linear Transformations, second edition, Daniel T. Finkbeiner, II. W.H.Freeman and Company, 1965.
Chapters on Complex numbers from Algebra for College Students, Ross R. Middlemiss. McGraw-Hill Book Company, Inc., 1953; and Plane Trigonometry with Tables, third edition, E.Richard Heineman. McGraw-Hill Book Company, Inc., 1964.
Statistics and Data Analysis in Geology, John C. Davis. John Wiley & Sons, 1973.
Introduction to Statistical Analysis, third edition, Wilfrid J. Dixon, Frank J. Massey, Jr. McGraw-Hill Book Company, 1969.
Applied Regression Analysis, second edition, Norman Draper, Harry Smith. John Wiley & Sons, 1981.
Multivariate Data Analysis, William W. Cooley, Paul R. Lohnes.
John Wiley & Sons, 1971
algebraic form of a complex number is defined by: (a + bi) where a is the real part, b is the imaginary part, and i is the square root of -1.
amplitude of a complex number (a + bi) is the angle theta equal arctan (b/a). Theta is used in the polar form.
ComplexNumbers - subclass of Magnitude containing common methods for its subclasses, ComplexNumber and ComplexPolarForm.
ComplexNumber class represents and manipulates the algebraic form of a complex number.
ComplexPolarForm class represents and manipulates the polar form of a complex number.
LowerTriangularMatrix class is a two-dimensional symmetrical matrix where only the lower half is stored. Dimensions are equal (n @ n) and total size equals (n * (n-1)) / 2+n.
LuMatrix class is a special form of SquareMatrix where the contents is the lu - decomposition of a matrix. Used to find the inverse, determinant, solve for a vector, and solve for a matrix.
Matrices class groups all matrices of two or more dimensions.
Matrices2Dimensional class groups matrices with two dimensions. Matrix class is the main class for matrices with 2 dimensions.
MatrixND class groups together matrices with 3 or more dimensions.
modulus of a complex number (a + bi) is also known as its absolute value. modulus equals square root of (a*a + b*b).
polar form of complex number is defined by: r(cos(theta) + i sin(theta)), where r is the modulus (positive number), theta is the amplitude (angle in radians), and i is the square root of -1.
real matrix means all the elements in the matrix are of class Number.
real symmetric matrix means that the matrix is real and symmetric.
scalar is an instance of Number (Integer, Fraction, or Float) or a ComplexNumber.
SquareMatrix is a subclass of Matrix where the number of rows equal the number of columns. The class exists to group a large number of matrix operations that are defined only for square matrices. System automatically creates a SquareMatrix when new: is sent to Matrix with an integer or point with x=y as an argument.
symmetrical matrix is dimensioned N by N and the elements a(i,j) = a(j,i) for all i=1..N and j=1..N except where i=j.
TriDiagonalMatrix class stores the three diagonals of a matrix. Elements not on one of the diagonals are considered to be zero. Dimensions are equal (n @ n) and total size equals 3 * n - 2.
trigonometric form of a complex number - see polar form.
Vector - a subclass of Array whose elements are scalars.
When a vector contains any complex numbers, certain methods are invalid
(e.g. #minimum, #maximum, #median which use inequality operators (<
>)).