Manuals/calci/PASCAL - Revision history

Devika: /* Examples */

2015-10-26T07:32:31Z

Examples

Devika: /* References */

2015-05-15T16:12:34Z

References

Devika at 18:42, 30 April 2015

2015-04-30T18:42:06Z

Devika: /* Description */

2015-04-30T18:32:52Z

Description

Devika at 18:30, 30 April 2015

2015-04-30T18:30:08Z

Abin: Created page with "
'''PASCAL'''

"

2015-04-30T05:28:41Z

Created page with "<div style="font-size:30px">'''PASCAL'''</div><br/>"

New page

<div style="font-size:30px">'''PASCAL'''</div><br/>

@@ Line 36: / Line 36: @@
 ==Examples==
-*1.MATRIX("pascal")
+*1.MATRIX("pascal") =1
 {| class="wikitable"
 |-
@@ Line 45: / Line 46: @@
 | 1 || 3 || 6
 |}
-*2.MATRIX("pascal",5)
+*3.MATRIX("pascal",5)
 {| class="wikitable"
 |-

← Older revision		Revision as of 16:12, 15 May 2015
Line 65:		Line 65:

	==References==		==References==
		+	*[http://en.wikipedia.org/wiki/Pascal_matrix Pascal Matrix]

← Older revision		Revision as of 18:42, 30 April 2015
Line 34:		Line 34:
	*Here MATRIX("pascal") is showing the pascal matrix of order 3.		*Here MATRIX("pascal") is showing the pascal matrix of order 3.
	*So users can change the order of the matrix also.		*So users can change the order of the matrix also.
		+
		+	==Examples==
		+	*1.MATRIX("pascal")
		+	{\| class="wikitable"
		+	\|-
		+	\| 1 \|\| 1 \|\| 1
		+	\|-
		+	\| 1 \|\| 2 \|\| 3
		+	\|-
		+	\| 1 \|\| 3 \|\| 6
		+	\|}
		+	*2.MATRIX("pascal",5)
		+	{\| class="wikitable"
		+	\|-
		+	\| 1 \|\| 1 \|\| 1 \|\| 1 \|\| 1
		+	\|-
		+	\| 1 \|\| 2 \|\| 3 \|\| 4 \|\| 5
		+	\|-
		+	\| 1 \|\| 3 \|\| 6 \|\| 10 \|\| 15
		+	\|-
		+	\| 1 \|\| 4 \|\| 10 \|\| 20 \|\| 35
		+	\|-
		+	\| 1 \|\| 5 \|\| 15 \|\| 35 \|\| 70
		+	\|}
		+
		+	==See Also==
		+	*[[Manuals/calci/ANTIDIAGONAL\| ANTIDIAGONAL]]
		+	*[[Manuals/calci/CONFERENCE\| CONFERENCE]]
		+	*[[Manuals/calci/TRIANGULAR\| TRIANGULAR]]
		+
		+	==References==

@@ Line 6: / Line 6: @@
 *The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
 *To obtain a pascal matrix there are three ways:  as either an upper-triangular matrix(U), a lower-triangular matrix(L), or a symmetric matrix(S).
-*Example for these matrices are:
+*Example for these matrices are
 <math>L_4 =\begin{pmatrix}
 & 0 & 0 & 0 \\

← Older revision		Revision as of 18:30, 30 April 2015
Line 1:		Line 1:
−	<div style="font-size:30px">'''PASCAL'''</div><br/>	+	<div style="font-size:30px">'''MATRIX("PASCAL",order)'''</div><br/>
		+	*<math>order</math> is the size of the Pascal matrix.
		+
		+	==Description==
		+	*This function returns the matrix of any order with the property of Pascal.
		+	*The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements.
		+	*To obtain a pascal matrix there are three ways: as either an upper-triangular matrix(U), a lower-triangular matrix(L), or a symmetric matrix(S).
		+	*Example for these matrices are:
		+	<math>L_4 =\begin{pmatrix}
		+	54 & 0 & 0 & 0 \\
		+	20 & 34 & 0 & 0 \\
		+	57 & 89 & -70 & 0 \\
		+	71 & -4 & -52 & 72 \\
		+	\end{pmatrix}</math>
		+	<math>U_4 =\begin{pmatrix}
		+	64 & 22 & -91 & -86 \\
		+	0 & 61 & 62 & -62 \\
		+	0 & 0 & 30 & -81 \\
		+	0 & 0 & 0 & -61 \\
		+	\end{pmatrix}</math>
		+	<math>S_4 =\begin{pmatrix}
		+	41 & 74 & 15 & -47 \\
		+	74 & -16 & 37 & 97 \\
		+	15 & 37 & 24 & -88 \\
		+	-47 & 97 & -88 & -69 \\
		+	\end{pmatrix}</math>
		+	*The amazing relationship of these matrices are:<math>S_n = L_nU_n</math>.
		+	*And its determinants also 1.i.e.,<math> \|S_n\|=\|L_n\|=\|U_n\|=1 </math>
		+	*The Pascal matrix can actually be constructed by taking the matrix exponential of a special subdiagonal or superdiagonal matrix.
		+	*The elements of the symmetric Pascal matrix are the binomial coefficients, i.e.
		+	<math>S_{ij} = {n \choose r} = \frac{n!}{r!(n-r)!},</math>, where n=i+j, r=i.
		+	*In other words,
		+	<math>S_{ij} = _{i+j}{C}_{i} = \frac{(i+j)!}{(i)!(j)!}</math>.
		+	*Here MATRIX("pascal") is showing the pascal matrix of order 3.
		+	*So users can change the order of the matrix also.