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DLALN2(3F)							    DLALN2(3F)


NAME    [Toc]    [Back]

     DLALN2 - solve a system of	the form (ca A - w D ) X = s B or (ca A' - w
     D)	X = s B	with possible scaling ("s") and	perturbation of	A

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	DLALN2(	LTRANS,	NA, NW,	SMIN, CA, A, LDA, D1, D2, B, LDB, WR,
			WI, X, LDX, SCALE, XNORM, INFO )

	 LOGICAL	LTRANS

	 INTEGER	INFO, LDA, LDB,	LDX, NA, NW

	 DOUBLE		PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM

	 DOUBLE		PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )

PURPOSE    [Toc]    [Back]

     DLALN2 solves a system of the form	 (ca A - w D ) X = s B or (ca A' - w
     D)	X = s B	  with possible	scaling	("s") and perturbation of A.  (A'
     means A-transpose.)

     A is an NA	x NA real matrix, ca is	a real scalar, D is an NA x NA real
     diagonal matrix, w	is a real or complex value, and	X and B	are NA x 1
     matrices -- real if w is real, complex if w is complex.  NA may be	1 or
     2.

     If	w is complex, X	and B are represented as NA x 2	matrices, the first
     column of each being the real part	and the	second being the imaginary
     part.

     "s" is a scaling factor (.LE. 1), computed	by DLALN2, which is so chosen
     that X can	be computed without overflow.  X is further scaled if
     necessary to assure that norm(ca A	- w D)*norm(X) is less than overflow.

     If	both singular values of	(ca A -	w D) are less than SMIN, SMIN*identity
     will be used instead of (ca A - w D).  If only one	singular value is less
     than SMIN,	one element of (ca A - w D) will be perturbed enough to	make
     the smallest singular value roughly SMIN.	If both	singular values	are at
     least SMIN, (ca A - w D) will not be perturbed.  In any case, the
     perturbation will be at most some small multiple of max( SMIN,
     ulp*norm(ca A - w D) ).  The singular values are computed by infinitynorm
 approximations, and thus will	only be	correct	to a factor of 2 or
     so.

     Note: all input quantities	are assumed to be smaller than overflow	by a
     reasonable	factor.	 (See BIGNUM.)

ARGUMENTS    [Toc]    [Back]

     LTRANS  (input) LOGICAL
	     =.TRUE.:  A-transpose will	be used.
	     =.FALSE.: A will be used (not transposed.)



									Page 1






DLALN2(3F)							    DLALN2(3F)



     NA	     (input) INTEGER
	     The size of the matrix A.	It may (only) be 1 or 2.

     NW	     (input) INTEGER
	     1 if "w" is real, 2 if "w"	is complex.  It	may only be 1 or 2.

     SMIN    (input) DOUBLE PRECISION
	     The desired lower bound on	the singular values of A.  This	should
	     be	a safe distance	away from underflow or overflow, say, between
	     (underflow/machine	precision) and	(machine precision * overflow
	     ).	 (See BIGNUM and ULP.)

     CA	     (input) DOUBLE PRECISION
	     The coefficient c,	which A	is multiplied by.

     A	     (input) DOUBLE PRECISION array, dimension (LDA,NA)
	     The NA x NA matrix	A.

     LDA     (input) INTEGER
	     The leading dimension of A.  It must be at	least NA.

     D1	     (input) DOUBLE PRECISION
	     The 1,1 element in	the diagonal matrix D.

     D2	     (input) DOUBLE PRECISION
	     The 2,2 element in	the diagonal matrix D.	Not used if NW=1.

     B	     (input) DOUBLE PRECISION array, dimension (LDB,NW)
	     The NA x NW matrix	B (right-hand side).  If NW=2 ("w" is
	     complex), column 1	contains the real part of B and	column 2
	     contains the imaginary part.

     LDB     (input) INTEGER
	     The leading dimension of B.  It must be at	least NA.

     WR	     (input) DOUBLE PRECISION
	     The real part of the scalar "w".

     WI	     (input) DOUBLE PRECISION
	     The imaginary part	of the scalar "w".  Not	used if	NW=1.

     X	     (output) DOUBLE PRECISION array, dimension	(LDX,NW)
	     The NA x NW matrix	X (unknowns), as computed by DLALN2.  If NW=2
	     ("w" is complex), on exit,	column 1 will contain the real part of
	     X and column 2 will contain the imaginary part.

     LDX     (input) INTEGER
	     The leading dimension of X.  It must be at	least NA.

     SCALE   (output) DOUBLE PRECISION
	     The scale factor that B must be multiplied	by to insure that
	     overflow does not occur when computing X.	Thus, (ca A - w	D) X



									Page 2






DLALN2(3F)							    DLALN2(3F)



	     will be SCALE*B, not B (ignoring perturbations of A.)  It will be
	     at	most 1.

     XNORM   (output) DOUBLE PRECISION
	     The infinity-norm of X, when X is regarded	as an NA x NW real
	     matrix.

     INFO    (output) INTEGER
	     An	error flag.  It	will be	set to zero if no error	occurs,	a
	     negative number if	an argument is in error, or a positive number
	     if	 ca A -	w D  had to be perturbed.  The possible	values are:
	     = 0: No error occurred, and (ca A - w D) did not have to be
	     perturbed.	 = 1: (ca A - w	D) had to be perturbed to make its
	     smallest (or only)	singular value greater than SMIN.  NOTE: In
	     the interests of speed, this routine does not check the inputs
	     for errors.
DLALN2(3F)							    DLALN2(3F)


NAME    [Toc]    [Back]

     DLALN2 - solve a system of	the form (ca A - w D ) X = s B or (ca A' - w
     D)	X = s B	with possible scaling ("s") and	perturbation of	A

SYNOPSIS    [Toc]    [Back]

     SUBROUTINE	DLALN2(	LTRANS,	NA, NW,	SMIN, CA, A, LDA, D1, D2, B, LDB, WR,
			WI, X, LDX, SCALE, XNORM, INFO )

	 LOGICAL	LTRANS

	 INTEGER	INFO, LDA, LDB,	LDX, NA, NW

	 DOUBLE		PRECISION CA, D1, D2, SCALE, SMIN, WI, WR, XNORM

	 DOUBLE		PRECISION A( LDA, * ), B( LDB, * ), X( LDX, * )

PURPOSE    [Toc]    [Back]

     DLALN2 solves a system of the form	 (ca A - w D ) X = s B or (ca A' - w
     D)	X = s B	  with possible	scaling	("s") and perturbation of A.  (A'
     means A-transpose.)

     A is an NA	x NA real matrix, ca is	a real scalar, D is an NA x NA real
     diagonal matrix, w	is a real or complex value, and	X and B	are NA x 1
     matrices -- real if w is real, complex if w is complex.  NA may be	1 or
     2.

     If	w is complex, X	and B are represented as NA x 2	matrices, the first
     column of each being the real part	and the	second being the imaginary
     part.

     "s" is a scaling factor (.LE. 1), computed	by DLALN2, which is so chosen
     that X can	be computed without overflow.  X is further scaled if
     necessary to assure that norm(ca A	- w D)*norm(X) is less than overflow.

     If	both singular values of	(ca A -	w D) are less than SMIN, SMIN*identity
     will be used instead of (ca A - w D).  If only one	singular value is less
     than SMIN,	one element of (ca A - w D) will be perturbed enough to	make
     the smallest singular value roughly SMIN.	If both	singular values	are at
     least SMIN, (ca A - w D) will not be perturbed.  In any case, the
     perturbation will be at most some small multiple of max( SMIN,
     ulp*norm(ca A - w D) ).  The singular values are computed by infinitynorm
 approximations, and thus will	only be	correct	to a factor of 2 or
     so.

     Note: all input quantities	are assumed to be smaller than overflow	by a
     reasonable	factor.	 (See BIGNUM.)

ARGUMENTS    [Toc]    [Back]

     LTRANS  (input) LOGICAL
	     =.TRUE.:  A-transpose will	be used.
	     =.FALSE.: A will be used (not transposed.)



									Page 1






DLALN2(3F)							    DLALN2(3F)



     NA	     (input) INTEGER
	     The size of the matrix A.	It may (only) be 1 or 2.

     NW	     (input) INTEGER
	     1 if "w" is real, 2 if "w"	is complex.  It	may only be 1 or 2.

     SMIN    (input) DOUBLE PRECISION
	     The desired lower bound on	the singular values of A.  This	should
	     be	a safe distance	away from underflow or overflow, say, between
	     (underflow/machine	precision) and	(machine precision * overflow
	     ).	 (See BIGNUM and ULP.)

     CA	     (input) DOUBLE PRECISION
	     The coefficient c,	which A	is multiplied by.

     A	     (input) DOUBLE PRECISION array, dimension (LDA,NA)
	     The NA x NA matrix	A.

     LDA     (input) INTEGER
	     The leading dimension of A.  It must be at	least NA.

     D1	     (input) DOUBLE PRECISION
	     The 1,1 element in	the diagonal matrix D.

     D2	     (input) DOUBLE PRECISION
	     The 2,2 element in	the diagonal matrix D.	Not used if NW=1.

     B	     (input) DOUBLE PRECISION array, dimension (LDB,NW)
	     The NA x NW matrix	B (right-hand side).  If NW=2 ("w" is
	     complex), column 1	contains the real part of B and	column 2
	     contains the imaginary part.

     LDB     (input) INTEGER
	     The leading dimension of B.  It must be at	least NA.

     WR	     (input) DOUBLE PRECISION
	     The real part of the scalar "w".

     WI	     (input) DOUBLE PRECISION
	     The imaginary part	of the scalar "w".  Not	used if	NW=1.

     X	     (output) DOUBLE PRECISION array, dimension	(LDX,NW)
	     The NA x NW matrix	X (unknowns), as computed by DLALN2.  If NW=2
	     ("w" is complex), on exit,	column 1 will contain the real part of
	     X and column 2 will contain the imaginary part.

     LDX     (input) INTEGER
	     The leading dimension of X.  It must be at	least NA.

     SCALE   (output) DOUBLE PRECISION
	     The scale factor that B must be multiplied	by to insure that
	     overflow does not occur when computing X.	Thus, (ca A - w	D) X



									Page 2






DLALN2(3F)							    DLALN2(3F)



	     will be SCALE*B, not B (ignoring perturbations of A.)  It will be
	     at	most 1.

     XNORM   (output) DOUBLE PRECISION
	     The infinity-norm of X, when X is regarded	as an NA x NW real
	     matrix.

     INFO    (output) INTEGER
	     An	error flag.  It	will be	set to zero if no error	occurs,	a
	     negative number if	an argument is in error, or a positive number
	     if	 ca A -	w D  had to be perturbed.  The possible	values are:
	     = 0: No error occurred, and (ca A - w D) did not have to be
	     perturbed.	 = 1: (ca A - w	D) had to be perturbed to make its
	     smallest (or only)	singular value greater than SMIN.  NOTE: In
	     the interests of speed, this routine does not check the inputs
	     for errors.


									PPPPaaaaggggeeee 3333
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