Bancroft Method - Revision history

Filipe.Pelica at 11:12, 7 July 2014

2014-07-07T11:12:05Z

← Older revision		Revision as of 11:12, 7 July 2014
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	The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions, ~~one of them is~~ the searched solution		The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions, that introduced in expression (10) provides the searched solution:

	::<math>		::<math>

2014-07-07T11:09:38Z

Reverted edits by Filipe.Pelica (talk) to last revision by Binbin.Wang

← Older revision		Revision as of 11:09, 7 July 2014
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	The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions~~. Introducing~~ one of ~~the solutions in expression (10) one obtains~~ the searched solution:		The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions, one of them is the searched solution

	::<math>		::<math>

2014-07-07T11:09:00Z

← Older revision		Revision as of 11:09, 7 July 2014
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	The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions ~~that introduced~~ in expression (10) ~~provides~~ the searched solution:		The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions. Introducing one of the solutions in expression (10) one obtains the searched solution:

	::<math>		::<math>

2014-07-07T11:06:33Z

← Older revision		Revision as of 11:06, 7 July 2014
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	The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions~~, one of them is~~ the searched solution		The previous expression (13) is a quadratic equation in <math>\Lambda</math> (note that matrix <math>{\mathbf B}</math> and the vector <math>\mathbf a </math> are also known) and provides two solutions that introduced in expression (10) provides the searched solution:

	::<math>		::<math>

2013-05-29T15:48:54Z

Raising and resolution

← Older revision		Revision as of 15:48, 29 May 2013
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	Developing the previous equation (4), it follows:		Developing the previous equation (4), it follows:

	::<math>\left[{x^j}^2+{y^j}^2+{z^j}^2-{PR^j}^2 \right]-2 \left[x^j x+y^j y+z^j z-{PR^~~j(c~~\,\delta t)} \; \right] + \left[x^2+y^2+z^2-(c\,\delta t)^2 \right]=0 \qquad \mbox{(5)}</math>		::<math>\left[{x^j}^2+{y^j}^2+{z^j}^2-{PR^j}^2 \right]-2 \left[x^j x+y^j y+z^j z-{PR^jc\,\delta t} \; \right] + \left[x^2+y^2+z^2-(c\,\delta t)^2 \right]=0 \qquad \mbox{(5)}</math>

2013-05-29T15:48:25Z

Raising and resolution

← Older revision		Revision as of 15:48, 29 May 2013
Line 45:		Line 45:
	Developing the previous equation (4), it follows:		Developing the previous equation (4), it follows:

	::<math>\left[{x^j}^2+{y^j}^2+{z^j}^2-{PR^j}^2 \right]-2 \left[x^j x+y^j y+z^j z-{PR^j} \; \right] + \left[x^2+y^2+z^2-(c\,\delta t)^2 \right]=0 \qquad \mbox{(5)}</math>		::<math>\left[{x^j}^2+{y^j}^2+{z^j}^2-{PR^j}^2 \right]-2 \left[x^j x+y^j y+z^j z-{PR^j(c\,\delta t)} \; \right] + \left[x^2+y^2+z^2-(c\,\delta t)^2 \right]=0 \qquad \mbox{(5)}</math>


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	The other solution is far from the earth surface.		The other solution is far from the earth surface.



	==Generalisation to the case of <math>n</math>-measurements:==		==Generalisation to the case of <math>n</math>-measurements:==

2013-01-13T11:44:49Z

← Older revision		Revision as of 11:44, 13 January 2013
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	::<math>		::<math>
	\begin{array}{r}		\begin{array}{r}
	R^j-D^j\simeq \sqrt{(x-x^j)^2+(y-y^j)^2+(z-z^j)^2}+c\,\delta t\\[0.3cm]		R^j-D^j\simeq \sqrt{(x^j-x)^2+(y^j-y)^2+(z^j-z)^2}+c\,\delta t\\[0.3cm]
	j=1,2,...,n~~~~ (n \geq 4)\\		j=1,2,...,n~~~~ (n \geq 4)\\
	\end{array}		\end{array}
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	::<math>		::<math>
	PR^j = \sqrt{(x-x^j)^2+(y-y^j)^2+(z-z^j)^2}+c \, \delta t		PR^j = \sqrt{(x^j-x)^2+(y^j-y)^2+(z^j-z)^2}+c \, \delta t
	\qquad \mbox{(4)}</math>		\qquad \mbox{(4)}</math>

2012-02-23T11:20:42Z

2012-02-02T12:33:23Z

← Older revision		Revision as of 12:33, 2 February 2012
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	Then, calling <math>{\mathbf r}=[x,y,z]^T</math> and considering the inner product of		Then, calling <math>{\mathbf r}=[x,y,z]^T</math> and considering the inner product of
	Lorentz <ref group=~~“footnotes”~~>		Lorentz <ref group="footnotes">
	<math>		<math>
	\left \langle{\mathbf a},{\mathbf b}\right \rangle={\mathbf a}^{t} \; {\mathbf M} \; {\mathbf b}=		\left \langle{\mathbf a},{\mathbf b}\right \rangle={\mathbf a}^{t} \; {\mathbf M} \; {\mathbf b}=

2012-02-02T12:28:21Z

← Older revision		Revision as of 12:28, 2 February 2012
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	==Notes==		==Notes==
	<references group="footnotes"/ >		<references group="footnotes"/>

	[[Category:Fundamentals]]		[[Category:Fundamentals]]
	[[Category:GNSS Measurements Modelling]]		[[Category:GNSS Measurements Modelling]]