1 SUBROUTINE DLANV2( A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN )
2 *
3 * -- LAPACK auxiliary routine (version 3.2.2) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * June 2010
7 *
8 * .. Scalar Arguments ..
9 DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
10 * ..
11 *
12 * Purpose
13 * =======
14 *
15 * DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
16 * matrix in standard form:
17 *
18 * [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
19 * [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
20 *
21 * where either
22 * 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
23 * 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
24 * conjugate eigenvalues.
25 *
26 * Arguments
27 * =========
28 *
29 * A (input/output) DOUBLE PRECISION
30 * B (input/output) DOUBLE PRECISION
31 * C (input/output) DOUBLE PRECISION
32 * D (input/output) DOUBLE PRECISION
33 * On entry, the elements of the input matrix.
34 * On exit, they are overwritten by the elements of the
35 * standardised Schur form.
36 *
37 * RT1R (output) DOUBLE PRECISION
38 * RT1I (output) DOUBLE PRECISION
39 * RT2R (output) DOUBLE PRECISION
40 * RT2I (output) DOUBLE PRECISION
41 * The real and imaginary parts of the eigenvalues. If the
42 * eigenvalues are a complex conjugate pair, RT1I > 0.
43 *
44 * CS (output) DOUBLE PRECISION
45 * SN (output) DOUBLE PRECISION
46 * Parameters of the rotation matrix.
47 *
48 * Further Details
49 * ===============
50 *
51 * Modified by V. Sima, Research Institute for Informatics, Bucharest,
52 * Romania, to reduce the risk of cancellation errors,
53 * when computing real eigenvalues, and to ensure, if possible, that
54 * abs(RT1R) >= abs(RT2R).
55 *
56 * =====================================================================
57 *
58 * .. Parameters ..
59 DOUBLE PRECISION ZERO, HALF, ONE
60 PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
61 DOUBLE PRECISION MULTPL
62 PARAMETER ( MULTPL = 4.0D+0 )
63 * ..
64 * .. Local Scalars ..
65 DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
66 $ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
67 * ..
68 * .. External Functions ..
69 DOUBLE PRECISION DLAMCH, DLAPY2
70 EXTERNAL DLAMCH, DLAPY2
71 * ..
72 * .. Intrinsic Functions ..
73 INTRINSIC ABS, MAX, MIN, SIGN, SQRT
74 * ..
75 * .. Executable Statements ..
76 *
77 EPS = DLAMCH( 'P' )
78 IF( C.EQ.ZERO ) THEN
79 CS = ONE
80 SN = ZERO
81 GO TO 10
82 *
83 ELSE IF( B.EQ.ZERO ) THEN
84 *
85 * Swap rows and columns
86 *
87 CS = ZERO
88 SN = ONE
89 TEMP = D
90 D = A
91 A = TEMP
92 B = -C
93 C = ZERO
94 GO TO 10
95 ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
96 $ THEN
97 CS = ONE
98 SN = ZERO
99 GO TO 10
100 ELSE
101 *
102 TEMP = A - D
103 P = HALF*TEMP
104 BCMAX = MAX( ABS( B ), ABS( C ) )
105 BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
106 SCALE = MAX( ABS( P ), BCMAX )
107 Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
108 *
109 * If Z is of the order of the machine accuracy, postpone the
110 * decision on the nature of eigenvalues
111 *
112 IF( Z.GE.MULTPL*EPS ) THEN
113 *
114 * Real eigenvalues. Compute A and D.
115 *
116 Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
117 A = D + Z
118 D = D - ( BCMAX / Z )*BCMIS
119 *
120 * Compute B and the rotation matrix
121 *
122 TAU = DLAPY2( C, Z )
123 CS = Z / TAU
124 SN = C / TAU
125 B = B - C
126 C = ZERO
127 ELSE
128 *
129 * Complex eigenvalues, or real (almost) equal eigenvalues.
130 * Make diagonal elements equal.
131 *
132 SIGMA = B + C
133 TAU = DLAPY2( SIGMA, TEMP )
134 CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
135 SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
136 *
137 * Compute [ AA BB ] = [ A B ] [ CS -SN ]
138 * [ CC DD ] [ C D ] [ SN CS ]
139 *
140 AA = A*CS + B*SN
141 BB = -A*SN + B*CS
142 CC = C*CS + D*SN
143 DD = -C*SN + D*CS
144 *
145 * Compute [ A B ] = [ CS SN ] [ AA BB ]
146 * [ C D ] [-SN CS ] [ CC DD ]
147 *
148 A = AA*CS + CC*SN
149 B = BB*CS + DD*SN
150 C = -AA*SN + CC*CS
151 D = -BB*SN + DD*CS
152 *
153 TEMP = HALF*( A+D )
154 A = TEMP
155 D = TEMP
156 *
157 IF( C.NE.ZERO ) THEN
158 IF( B.NE.ZERO ) THEN
159 IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
160 *
161 * Real eigenvalues: reduce to upper triangular form
162 *
163 SAB = SQRT( ABS( B ) )
164 SAC = SQRT( ABS( C ) )
165 P = SIGN( SAB*SAC, C )
166 TAU = ONE / SQRT( ABS( B+C ) )
167 A = TEMP + P
168 D = TEMP - P
169 B = B - C
170 C = ZERO
171 CS1 = SAB*TAU
172 SN1 = SAC*TAU
173 TEMP = CS*CS1 - SN*SN1
174 SN = CS*SN1 + SN*CS1
175 CS = TEMP
176 END IF
177 ELSE
178 B = -C
179 C = ZERO
180 TEMP = CS
181 CS = -SN
182 SN = TEMP
183 END IF
184 END IF
185 END IF
186 *
187 END IF
188 *
189 10 CONTINUE
190 *
191 * Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
192 *
193 RT1R = A
194 RT2R = D
195 IF( C.EQ.ZERO ) THEN
196 RT1I = ZERO
197 RT2I = ZERO
198 ELSE
199 RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
200 RT2I = -RT1I
201 END IF
202 RETURN
203 *
204 * End of DLANV2
205 *
206 END
2 *
3 * -- LAPACK auxiliary routine (version 3.2.2) --
4 * -- LAPACK is a software package provided by Univ. of Tennessee, --
5 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6 * June 2010
7 *
8 * .. Scalar Arguments ..
9 DOUBLE PRECISION A, B, C, CS, D, RT1I, RT1R, RT2I, RT2R, SN
10 * ..
11 *
12 * Purpose
13 * =======
14 *
15 * DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
16 * matrix in standard form:
17 *
18 * [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
19 * [ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
20 *
21 * where either
22 * 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
23 * 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
24 * conjugate eigenvalues.
25 *
26 * Arguments
27 * =========
28 *
29 * A (input/output) DOUBLE PRECISION
30 * B (input/output) DOUBLE PRECISION
31 * C (input/output) DOUBLE PRECISION
32 * D (input/output) DOUBLE PRECISION
33 * On entry, the elements of the input matrix.
34 * On exit, they are overwritten by the elements of the
35 * standardised Schur form.
36 *
37 * RT1R (output) DOUBLE PRECISION
38 * RT1I (output) DOUBLE PRECISION
39 * RT2R (output) DOUBLE PRECISION
40 * RT2I (output) DOUBLE PRECISION
41 * The real and imaginary parts of the eigenvalues. If the
42 * eigenvalues are a complex conjugate pair, RT1I > 0.
43 *
44 * CS (output) DOUBLE PRECISION
45 * SN (output) DOUBLE PRECISION
46 * Parameters of the rotation matrix.
47 *
48 * Further Details
49 * ===============
50 *
51 * Modified by V. Sima, Research Institute for Informatics, Bucharest,
52 * Romania, to reduce the risk of cancellation errors,
53 * when computing real eigenvalues, and to ensure, if possible, that
54 * abs(RT1R) >= abs(RT2R).
55 *
56 * =====================================================================
57 *
58 * .. Parameters ..
59 DOUBLE PRECISION ZERO, HALF, ONE
60 PARAMETER ( ZERO = 0.0D+0, HALF = 0.5D+0, ONE = 1.0D+0 )
61 DOUBLE PRECISION MULTPL
62 PARAMETER ( MULTPL = 4.0D+0 )
63 * ..
64 * .. Local Scalars ..
65 DOUBLE PRECISION AA, BB, BCMAX, BCMIS, CC, CS1, DD, EPS, P, SAB,
66 $ SAC, SCALE, SIGMA, SN1, TAU, TEMP, Z
67 * ..
68 * .. External Functions ..
69 DOUBLE PRECISION DLAMCH, DLAPY2
70 EXTERNAL DLAMCH, DLAPY2
71 * ..
72 * .. Intrinsic Functions ..
73 INTRINSIC ABS, MAX, MIN, SIGN, SQRT
74 * ..
75 * .. Executable Statements ..
76 *
77 EPS = DLAMCH( 'P' )
78 IF( C.EQ.ZERO ) THEN
79 CS = ONE
80 SN = ZERO
81 GO TO 10
82 *
83 ELSE IF( B.EQ.ZERO ) THEN
84 *
85 * Swap rows and columns
86 *
87 CS = ZERO
88 SN = ONE
89 TEMP = D
90 D = A
91 A = TEMP
92 B = -C
93 C = ZERO
94 GO TO 10
95 ELSE IF( ( A-D ).EQ.ZERO .AND. SIGN( ONE, B ).NE.SIGN( ONE, C ) )
96 $ THEN
97 CS = ONE
98 SN = ZERO
99 GO TO 10
100 ELSE
101 *
102 TEMP = A - D
103 P = HALF*TEMP
104 BCMAX = MAX( ABS( B ), ABS( C ) )
105 BCMIS = MIN( ABS( B ), ABS( C ) )*SIGN( ONE, B )*SIGN( ONE, C )
106 SCALE = MAX( ABS( P ), BCMAX )
107 Z = ( P / SCALE )*P + ( BCMAX / SCALE )*BCMIS
108 *
109 * If Z is of the order of the machine accuracy, postpone the
110 * decision on the nature of eigenvalues
111 *
112 IF( Z.GE.MULTPL*EPS ) THEN
113 *
114 * Real eigenvalues. Compute A and D.
115 *
116 Z = P + SIGN( SQRT( SCALE )*SQRT( Z ), P )
117 A = D + Z
118 D = D - ( BCMAX / Z )*BCMIS
119 *
120 * Compute B and the rotation matrix
121 *
122 TAU = DLAPY2( C, Z )
123 CS = Z / TAU
124 SN = C / TAU
125 B = B - C
126 C = ZERO
127 ELSE
128 *
129 * Complex eigenvalues, or real (almost) equal eigenvalues.
130 * Make diagonal elements equal.
131 *
132 SIGMA = B + C
133 TAU = DLAPY2( SIGMA, TEMP )
134 CS = SQRT( HALF*( ONE+ABS( SIGMA ) / TAU ) )
135 SN = -( P / ( TAU*CS ) )*SIGN( ONE, SIGMA )
136 *
137 * Compute [ AA BB ] = [ A B ] [ CS -SN ]
138 * [ CC DD ] [ C D ] [ SN CS ]
139 *
140 AA = A*CS + B*SN
141 BB = -A*SN + B*CS
142 CC = C*CS + D*SN
143 DD = -C*SN + D*CS
144 *
145 * Compute [ A B ] = [ CS SN ] [ AA BB ]
146 * [ C D ] [-SN CS ] [ CC DD ]
147 *
148 A = AA*CS + CC*SN
149 B = BB*CS + DD*SN
150 C = -AA*SN + CC*CS
151 D = -BB*SN + DD*CS
152 *
153 TEMP = HALF*( A+D )
154 A = TEMP
155 D = TEMP
156 *
157 IF( C.NE.ZERO ) THEN
158 IF( B.NE.ZERO ) THEN
159 IF( SIGN( ONE, B ).EQ.SIGN( ONE, C ) ) THEN
160 *
161 * Real eigenvalues: reduce to upper triangular form
162 *
163 SAB = SQRT( ABS( B ) )
164 SAC = SQRT( ABS( C ) )
165 P = SIGN( SAB*SAC, C )
166 TAU = ONE / SQRT( ABS( B+C ) )
167 A = TEMP + P
168 D = TEMP - P
169 B = B - C
170 C = ZERO
171 CS1 = SAB*TAU
172 SN1 = SAC*TAU
173 TEMP = CS*CS1 - SN*SN1
174 SN = CS*SN1 + SN*CS1
175 CS = TEMP
176 END IF
177 ELSE
178 B = -C
179 C = ZERO
180 TEMP = CS
181 CS = -SN
182 SN = TEMP
183 END IF
184 END IF
185 END IF
186 *
187 END IF
188 *
189 10 CONTINUE
190 *
191 * Store eigenvalues in (RT1R,RT1I) and (RT2R,RT2I).
192 *
193 RT1R = A
194 RT2R = D
195 IF( C.EQ.ZERO ) THEN
196 RT1I = ZERO
197 RT2I = ZERO
198 ELSE
199 RT1I = SQRT( ABS( B ) )*SQRT( ABS( C ) )
200 RT2I = -RT1I
201 END IF
202 RETURN
203 *
204 * End of DLANV2
205 *
206 END