Linear algebra

MATRIX |

[Features] To execute various
calculations of Linear
algebra.

[Format] MATRIX "Array1=
eigen(Array2)"

[Explanation]

Description
method:

matrix "VariableName = Linear algebra
Formula"

Right side is the various function names
exclusive for 'MATRIX'

and Array-variable name of
the data source in parentheses.

The format in the case of
Four-arithmetic-operations calculation,

"Array-variable1
Formula Array-variable2"

The variable name is Numeric
array variable without a parenthesis.

The case of which
left side is numerical value variable not Array:

->Determinant and rank.

The array variables to be used
must be declared at the beginning of the program in
advance.

If the size of array for substitute is different
than requested one, automatically resized.

(When different
scope in [Array variable] and ['MATRIX' command
execution],

since resizing is impossible, it become
an error)

*About Complex
numbers of input and output values, these
are not supported at this stage.

Matrix data are given by Array
variable.

(Think of array variables as
matrix)

If the columns count is [x] and
the rows count is [y], it declare as [ dim dv1(y,x) ].

If
declaring [ dim dv1(2) ] in 'Basic' Array variable,

3 variables of [0-2] will be
declared.

This is one larger size than the desired
size,

option base -1 (Please see the manual [OPTION
BASE] for details)

By writing this command, to make it to
allocate 2 data from [0-1].

This allocate become the same
number as general language, C or Java.

When dealing with
linear algebra,

be sure to write [option base -1] at
the beginning of the program.

When not writting, correct
calculation result isn't obtained.

About errors in linear
algebra.

If uncalculable error occurs in Linear algebra
calculation,

the program is not interrupted, it only
return error code to 'ERR' function.

Please check the
success or failure of calculation in the state of 'ERR'
function.

'ERR' function is reset to 0 at the beginning of
'MATRIX' command execution.

Linear algebra related
errors.

error 69:"Matrix error, uncalculable or
cofactor 0"

error 70:"Array size of matrix does not
fit"

Four arithmetic operations of Linear
algebra.

e.g

10 option base-1

20 dim dv1(2,2),dv2(2,2),dv3(2,2)

30 dv2(0,0)=4:dv2(0,1)=-2:dv2(1,0)=1:dv2(1,1)=1

40 dv3(0,0)=1:dv3(0,1)=2:dv3(1,0)=3:dv3(1,1)=4

50 matrix "dv1=dv2+dv3"

60 for y=0 to 1 : for x=0 to 1

70 print dv1(y,x);

80 next : print : next

As something defined already up to 40
line, to explained using an example

in which only
[line50 'MATRIX' parameter] is changed.

('matrix' next
line is result)

Addition, Subtraction: It must be the same
size in two calculation arrays.

Multiply: It must be the same size in [left
array column count] and [right array row count]

Division: The two arrays must be square matricx of the same size.

[Addition]

matrix
"dv1=dv2+dv3"

5 0

4 5

[Subtraction]

matrix
"dv1=dv2-dv3"

3 -4

-2 -3

[Multiply]

matrix
"dv1=dv2*dv3"

-2 0

4 6

[Division]

matrix
"dv1=dv2/dv3"

-7 5

5.5
-3.5

There is no division in linear
algebra,

it return the value obtained by multiplying
by

[Inverse matrix of division-matrix-dv3] from left
side of dv2.

--Internal function of 'MATRIX' command exclusive use--

e.g.
forEigenvalue

10 optionbase-1

20 dimdv1(2,2),dv2(2,2)

30 dv2(0,0)=4:dv2(0,1)=-2:dv2(1,0)=1:dv2(1,1)=1

40 matrix "dv1=eigen(dv2)"

50 for y=0 to 1 : for x=0 to 1

60 print dv1(y,x);

70 next : print : next

As something defined already up to 30
line, to explained using an example

in which only
[line40 'MATRIX' parameter] is changed.

('matrix' next
line is result)

[ DET ]

To
calculate Determinant from given Matrix.

matrix
"var=det(dv2)"

6

The left side
specifies the usual numeric variable, not an array.

The
array(dv2) that give the data must be a square matrix.

[ INV ]

To
calculate Inverse from given Matrix.

matrix
"dv1=inv(dv2)"

0.1666 0.333

-0.1666
0.666

The array(dv1) to substitute must be
a square matrix

of the same size as the array(dv2)
giving data.

If Determinant of the given matrix is 0,
Inverse matrix doesn't exist,

so it become ['ERR'
function=69] and it isn't calculated.

(Program dosn't
interruption)

Others that use the inverse matrix during
the calculation

(Division, Simultaneous
equations),

if the Determinant becomes 0, to become
['ERR' function=69].

[ EQUAT ]

To
find the solution of simultaneous equations with a matrix.

e.g. for Simultaneous equations.

x+2y=4

3x+5y=9

Data to prepare(2x3)

1 2 4

3 5 9

Program to find the solution of simultaneous equations
of matrix with 'EQUAT' function.

10 option base -1

20 dim dv1(2,1),dv2(2,3)

30 dv2(0,0)=1:dv2(0,1)=2:dv2(0,2)=4

40 dv2(1,0)=3:dv2(1,1)=5:dv2(1,2)=9

50 matrix "dv1=equat(dv2)"

60 for y=0 to 1

70 print dv1(y,0);

80 next

Output solution

-2

3

(x=-2,y=3)

*If
rank of matrix is less than the number of rows, the solution may notbefound.

When the right side of the expression
is all 0.

x-3y=0

2x-6y=0

Data in an array.

1 -3
0

2 -6 0

The mutual ratio between
each character expression.

9

3

If the right side is all
0,

the solution become impossible or
indetermination,

it cann't find a
solution.

But if it is indetermination, it's possible to
calculate

[The mutual ratio between each character
expression].

So in this case, the ratio is returned.(by
integer)

In this example,

to
verify with [substitute 9 for x],[substitute 3 for y]

It
becomes 0, same as the right side,

it turns out that
the result is right.

In the case of this ratio, if the
calculation is impossible,

it become ['ERR'
function=69], and the calculation is not performed.

About the size of the array
used.

The array(dv2) that give the equation
data,

it prepare columns count 1
larger than a square matrix.

The rightmost column will
enter the numbers on right side of the equation.

It set
size [columns count=1, rows count=same as array that give the
data]

that the
array be substitutioned the solution.

It become that the
solutions is lined up in a column vertical.

[ EIGEN ]

To
calculate Eigenvalue from given Matrix.

matrix
"dv1=eigen(dv2)"

3 0

0
2

The eigenvalues are arranged diagonally in the array as
in the example, and returned.

Because it doesn't supported
to a complex number,

If the eigenvalue is complex number,
unregular value is returned.

(It is not implemented that the judgment
way of whether solution is complex number, yet)

The array(dv1) to substitute
must be a square matrix

of the same size as the
array(dv2) giving data.

[ EVEC ]

To
calculate Eigenvector from given Matrix.

matrix
"dv1=evec(dv2)"

2 1

1
1

The
format of the returned array is this: (vertical
arrangement)

The Eigenvector of the 0th Eigenvalue is in
0th column,

The Eigenvector of the 1th Eigenvalue is
in 1th column...

The array(dv1) to substitute must be a
square matrix

of the same size as the array(dv2)
giving data.

[ TRANS ]

To
return Transpose matrix from given Matrix.

matrix
"dv1=trans(dv2)"

4 1

-2
1

The array(dv1) to substitute must be a square
matrix

of the same size as the array(dv2) giving
data.

[ RANK ]

To
return Rank from given Matrix.

matrix
"var=rank(dv2)"

2

The left side
specifies the usual numeric variable, not an array.

[ UNIT ]

To
return Unit matrix of specified number.

matrix
"dv1=unit(2)"

1 0

0
1

To specify the square size in a number of the
function.

[ INTCV ]

The
data in array is integer-ized.

Format:
intcv(ArrayName,ProcessingLine,MaxInspectionValue,DetermineDigits)

e.g.: matrix "dv1=intcv(dv2,-1,100,0.00001)"

It examine
whether [Each value in the given array] will be [Shaped integer
ratio],

If possible, Each value is converted to data in
the ratio of integer, and returned.

If it can't change to
integer, to return the array of original values as is.

('ERR' function=69, None interruption)

The range to be
integerized can be specified either [Whole of array] or [Specified
row].

To use it in this case. For
example,

Matrix question made by a person that have an
integer in the solution.

It try to make this be
integer-ized.

Left side:

Array(dv1) to substitute the result.

-Parameter in the
function-

ArrayName:

Array(dv2)
which try to be integer-ized.

ProcessingLine: (Specify
from 0-)

To specify which column to be
integer-ized.

If -1 is specified, everything in the
array is targeted.

MaxInspectionValue:

It
is investigated to which integral number

actually
corresponds to minimum value that be standard integer.

To
set the upper limit value of the survey.

If the survey
reaches the upper limit and there is no prospect of
integer-ized,

it is judged that integer-izing is
impossible.

('ERR' function=69, None
interruption)

The larger this value, the more likely it is
to be integer-ization,

but the processing load will
be large.

DetermineDigits:

The larger
digit numbers of a decimal(small number),

the
stricter judge criteria of integer-ized.

The smaller digit
numbers(big number) then roughly.

If the value is not well
known, please use the value of

[
intcv(ArrayName,-1,100,0.00001) ] like the example.

The
array(dv1) to substitute must be a square matrix

of
the same size as the array(dv2) giving data.

[ SETTING ]

This is function which do setting, not calculation.

matrix
"var=setting(ret,0)"

Various setting of Linear
algebraic.

To write the type of setting in the first
parameter of function.

(This word don't need to be
enclosed in double quotation)

-First
parameter-

[ret]

To set whether
the return value of Eigenvector is returned as a decimal or an
integer.

-Second parameter-

0:
Return value as a normalized decimal.

1: Return
value as an integer.(default)

The left side is a temporary
variable without meaning.

Please specify a normal
numeric variable, not an array.

-The
settings to decide number of iterations to convergence, judgment
digits

in calculating eigenvalue,
eigenvector.

[digit]

Setting the
digits to judge whether decimal converge to integers.

For
example, 'setting(digit,3)' [default=3] then,

the eigenvector '2.000567' is below the decimal point, and '0' continues 3
times,

it is judged to converge on '2' of
integer.

The larger the number, the stricter the
judgment.

(the following number of 'rep' times also
require a large number)

[rep]

The
number of iterations when finding eigenvalues by QR
decomposition.

[default]
setting(digit,3200)

The number of times is larger, the
more accurate value can be obtained.

---------------------------

How to use the sample "linear.bas"

---------------------------

When executed, the command selection screen appears.

Enter
the number.

1.input 2.save 3.load 4.calc

First, enter the data with '1.input'

To enter the size of
the row and column of the matrix.

size x? (Enter columns
count)

size y? (Enter rows count)

Let's enter the
size of '2x2' as an example.

(When equation data, to make the
columns count increase by 1)

Next enter the data.

4 -2

1 1

Let's
enter this matrix data.

The columns number are counted from the right
side.

0,0 ? 4

0,1 ? -2

1,0 ? 1

1,1 ? 1

Let's saves the input data. (to memo
the size of the matrix.)

Please select [ 2.save ]

Enter the
save number. To try saving to number 1.

No.1

It was saved by
pressing [Enter].

Let's actually see the calculation
result.

Please select [ 4.calc ]

Matrix data

(Rank)

(det)

(mult:squared)

displayed in this
order.

To tap the key to display more data.

(inv)

(Eigen)

(Evec)

This data will be displayed.

Finally, the loading procedure is explained.

Please select
[ 3.load ]

To enter the load number. Then try to load [No.1]
earlier.

No.1

It's necessary to input the matrix
size.

Enter the size you noted earlier.

size x? 2 (Enter columns
count)

size y? 2 (Enter rows count)

It is now
loaded.

It's possible to make calculation result indicate
variously

by choosing [4.calc] by the procedure earlier.