ct_methods Module Reference

Cayley transformation methods. More...

Functions/Subroutines
subroutine, public	ct_step_execute (cts_env)
	Performs Cayley transformation.
subroutine, public	analytic_line_search (a, b, c, d, minima, nmins)
	Finds real roots of a cubic equation ‍ a*x**3 + b*x**2 + c*x + d = 0 and returns only those roots for which the derivative is positive Step 0: Check the true order of the equation. Cubic, quadratic, linear? Step 1: Calculate p and q p = ( 3*c/a - (b/a)**2 ) / 3 q = ( 2*(b/a)**3 - 9*b*c/a/a + 27*d/a ) / 27 Step 2: Calculate discriminant D D = (p/3)**3 + (q/2)**2 Step 3: Depending on the sign of D, we follow different strategy. If D<0, three distinct real roots. If D=0, three real roots of which at least two are equal. If D>0, one real and two complex roots. Step 3a: For D>0 and D=0, Calculate u and v u = cubic_root(-q/2 + sqrt(D)) v = cubic_root(-q/2 - sqrt(D)) Find the three transformed roots y1 = u + v y2 = -(u+v)/2 + i (u-v)*sqrt(3)/2 y3 = -(u+v)/2 - i (u-v)*sqrt(3)/2 Step 3b Alternately, for D<0, a trigonometric formulation is more convenient y1 = 2 * sqrt(|p|/3) * cos(phi/3) y2 = -2 * sqrt(|p|/3) * cos((phi+pi)/3) y3 = -2 * sqrt(|p|/3) * cos((phi-pi)/3) where phi = acos(-q/2/sqrt(|p|**3/27)) pi = 3.141592654... Step 4 Find the real roots x = y - b/a/3 Step 5 Check the derivative and return only those real roots for which the derivative is positive.
subroutine, public	diagonalize_diagonal_blocks (matrix, c, e)
	Diagonalizes diagonal blocks of a symmetric dbcsr matrix and returs its eigenvectors.

Detailed Description

Cayley transformation methods.

History

2011.06 created [Rustam Z Khaliullin]

Author

Rustam Z Khaliullin

Function/Subroutine Documentation

ct_step_execute()

subroutine, public ct_methods::ct_step_execute

(

type(ct_step_env_type)

cts_env

)

Performs Cayley transformation.

Parameters

cts_env

...

History

2011.06 created [Rustam Z Khaliullin]

Author

Rustam Z Khaliullin

Definition at line 62 of file ct_methods.F.

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analytic_line_search()

subroutine, public ct_methods::analytic_line_search	(	real(kind=dp), intent(in)	a,
		real(kind=dp), intent(in)	b,
		real(kind=dp), intent(in)	c,
		real(kind=dp), intent(in)	d,
		real(kind=dp), dimension(3), intent(out)	minima,
		integer, intent(out)	nmins
	)

Finds real roots of a cubic equation

‍ a*x**3 + b*x**2 + c*x + d = 0 and returns only those roots for which the derivative is positive

Step 0: Check the true order of the equation. Cubic, quadratic, linear? Step 1: Calculate p and q p = ( 3*c/a - (b/a)**2 ) / 3 q = ( 2*(b/a)**3 - 9*b*c/a/a + 27*d/a ) / 27 Step 2: Calculate discriminant D D = (p/3)**3 + (q/2)**2 Step 3: Depending on the sign of D, we follow different strategy. If D<0, three distinct real roots. If D=0, three real roots of which at least two are equal. If D>0, one real and two complex roots. Step 3a: For D>0 and D=0, Calculate u and v u = cubic_root(-q/2 + sqrt(D)) v = cubic_root(-q/2 - sqrt(D)) Find the three transformed roots y1 = u + v y2 = -(u+v)/2 + i (u-v)*sqrt(3)/2 y3 = -(u+v)/2 - i (u-v)*sqrt(3)/2 Step 3b Alternately, for D<0, a trigonometric formulation is more convenient y1 = 2 * sqrt(|p|/3) * cos(phi/3) y2 = -2 * sqrt(|p|/3) * cos((phi+pi)/3) y3 = -2 * sqrt(|p|/3) * cos((phi-pi)/3) where phi = acos(-q/2/sqrt(|p|**3/27)) pi = 3.141592654... Step 4 Find the real roots x = y - b/a/3 Step 5 Check the derivative and return only those real roots for which the derivative is positive.

Parameters

a	...
b	...
c	...
d	...
minima	...
nmins	...

History

2011.06 created [Rustam Z Khaliullin]

Author

Rustam Z Khaliullin

Definition at line 1374 of file ct_methods.F.

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diagonalize_diagonal_blocks()

subroutine, public ct_methods::diagonalize_diagonal_blocks	(	type(dbcsr_type), intent(in)	matrix,
		type(dbcsr_type), intent(out)	c,
		type(dbcsr_type), intent(out), optional	e
	)

Diagonalizes diagonal blocks of a symmetric dbcsr matrix and returs its eigenvectors.

Parameters

matrix	...
c	...
e	...

History

2011.07 created [Rustam Z Khaliullin]

Author

Rustam Z Khaliullin

Definition at line 1497 of file ct_methods.F.

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