1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
|
*DECK ALQUAR
SUBROUTINE ALQUAR(A,ROOTS)
*
*-----------------------------------------------------------------------
*
*Purpose:
* compute the roots of the real quartic polynomial defined as
* A(1)+A(2)*Z + ... + A(5)*Z**4.
* NOTE: It is assumed that A(5) is non-zero. No test is made here.
*
*Author(s): A. H. Morris, W. L. Davis, A. Miller, and R. L. Carmichael
*
*Parameters: input
* A polynomial coefficients
*
*Parameters: output
* ROOTS complex roots
*
*-----------------------------------------------------------------------
*
IMPLICIT NONE
*----
* SUBROUTINE ARGUMENTS
*----
DOUBLE PRECISION, INTENT(IN) :: A(5)
COMPLEX*16, INTENT(OUT) :: ROOTS(4)
*----
* LOCAL VARIABLES
*----
INTEGER :: I, J
INTEGER,DIMENSION(1) :: K
DOUBLE PRECISION :: B, B2, C, D, E, H, P, Q, R, T, WORK, U, V,
> V1, V2, X, XX(3), Y, BQ, CQ, DQ, AA, BB
COMPLEX*16 :: W, AAA, BBB, SQ1, TEST, SQRTM3
DOUBLE PRECISION,DIMENSION(4) :: TEMP
PARAMETER (SQRTM3=(0.0,1.73205080756888))
*
IF(A(1)==0.0) THEN
IF(A(2).EQ.0.0) THEN
CQ=A(4)/A(5)
DQ=A(3)/A(5)
AAA=CQ*CQ-4.0D0*DQ
AAA=SQRT(AAA)
ROOTS(1)=0.0
ROOTS(2)=0.0
ROOTS(3)=-0.5D0*(CQ+AAA)
ROOTS(4)=-0.5D0*(CQ-AAA)
ELSE
BQ=A(4)/A(5)
CQ=A(3)/A(5)
DQ=A(2)/A(5)
AA=(3.0D0*CQ-BQ**2)/3.0D0
BB=(2.0D0*BQ**3-9.0D0*BQ*CQ+27.0D0*DQ)/27.0D0
SQ1=BB**2/4.0D0+AA**3/27.0D0
TEST=BB/2.0D0-SQRT(SQ1)
IF(DBLE(TEST).EQ.0.0) THEN
AAA=0.0D0
ELSE IF(DBLE(TEST).GT.0.0) THEN
AAA=-(TEST)**(1.0D0/3.0D0)
ELSE
AAA=(-TEST)**(1.0D0/3.0D0)
ENDIF
TEST=BB/2.0D0+SQRT(SQ1)
IF(DBLE(TEST).EQ.0.0) THEN
BBB=0.0D0
ELSE IF(DBLE(TEST).GT.0.0) THEN
BBB=-(TEST)**(1.0D0/3.0D0)
ELSE
BBB=(-TEST)**(1.0D0/3.0D0)
ENDIF
ROOTS(1)=0.0
ROOTS(2)=AAA+BBB-BQ/3.0D0
ROOTS(3)=-(AAA+BBB)/2.0D0+(AAA-BBB)*SQRTM3/2.0D0-BQ/3.0D0
ROOTS(4)=-(AAA+BBB)/2.0D0-(AAA-BBB)*SQRTM3/2.0D0-BQ/3.0D0
ENDIF
RETURN
ENDIF
*----
* Solve a quartic equation
*----
B = A(4)/(4.0D0*A(5))
C = A(3)/A(5)
D = A(2)/A(5)
E = A(1)/A(5)
B2 = B*B
P = 0.5D0*(C - 6.0D0*B2)
Q = D - 2.0D0*B*(C - 4.0D0*B2)
R = B2*(C - 3.0D0*B2) - B*D + E
*----
* Solve the resolvent cubic equation. the cubic has at least one
* nonnegative real root. if W1, W2, W3 are the roots of the cubic
* then the roots of the original equation are
* ROOTS = -B + CSQRT(W1) + CSQRT(W2) + CSQRT(W3)
* where the signs of the square roots are chosen so
* that CSQRT(W1) * CSQRT(W2) * CSQRT(W3) = -Q/8.
*----
TEMP(1) = -Q*Q/64.0D0
TEMP(2) = 0.25D0*(P*P - R)
TEMP(3) = P
TEMP(4) = 1.0D0
BQ=TEMP(3)
CQ=TEMP(2)
DQ=TEMP(1)
AA=(3.0D0*CQ-BQ**2)/3.0D0
BB=(2.0D0*BQ**3-9.0D0*BQ*CQ+27.0D0*DQ)/27.0D0
SQ1=BB**2/4.0D0+AA**3/27.0D0
TEST=BB/2.0D0-SQRT(SQ1)
IF(DBLE(TEST).EQ.0.0) THEN
AAA=0.0D0
ELSE IF(DBLE(TEST).GT.0.0) THEN
AAA=-(TEST)**(1.0D0/3.0D0)
ELSE
AAA=(-TEST)**(1.0D0/3.0D0)
ENDIF
TEST=BB/2.0D0+SQRT(SQ1)
IF(DBLE(TEST).EQ.0.0) THEN
BBB=0.0D0
ELSE IF(DBLE(TEST).GT.0.0) THEN
BBB=-(TEST)**(1.0D0/3.0D0)
ELSE
BBB=(-TEST)**(1.0D0/3.0D0)
ENDIF
ROOTS(1)=AAA+BBB-BQ/3.0D0
ROOTS(2)=-(AAA+BBB)/2.0D0+(AAA-BBB)*SQRTM3/2.0D0-BQ/3.0D0
ROOTS(3)=-(AAA+BBB)/2.0D0-(AAA-BBB)*SQRTM3/2.0D0-BQ/3.0D0
IF(AIMAG(ROOTS(2)).NE.0.0D0) GO TO 60
*----
* The resolvent cubic has only real roots.
* Reorder the roots in increasing order.
*----
XX(1) = DBLE(ROOTS(1))
XX(2) = DBLE(ROOTS(2))
XX(3) = DBLE(ROOTS(3))
DO 25 J=2,3
X=XX(J)
DO 10 I=J-1,1,-1
IF(XX(I).LE.X) GOTO 20
XX(I+1)=XX(I)
10 CONTINUE
I=0
20 XX(I+1)=X
25 CONTINUE
U = 0.0D0
IF(XX(3).GT.0.0D0) U = SQRT(XX(3))
IF(XX(2).LE.0.0D0) GO TO 41
IF(XX(1).GE.0.0D0) GO TO 30
IF(ABS(XX(1)).GT.XX(2)) GO TO 40
XX(1) = 0.0D0
30 XX(1) = SQRT(XX(1))
XX(2) = SQRT(XX(2))
IF(Q.GT.0.0D0) XX(1) = -XX(1)
TEMP(1) = (( XX(1) + XX(2)) + U) - B
TEMP(2) = ((-XX(1) - XX(2)) + U) - B
TEMP(3) = (( XX(1) - XX(2)) - U) - B
TEMP(4) = ((-XX(1) + XX(2)) - U) - B
DO J=1,3
K=MINLOC(TEMP(J:))
IF(J.NE.K(1)) THEN
WORK = TEMP(J)
TEMP(J) = TEMP(K(1))
TEMP(K(1)) = WORK
ENDIF
ENDDO
IF(ABS(TEMP(1)).GE.0.1D0*ABS(TEMP(4))) GO TO 31
T = TEMP(2)*TEMP(3)*TEMP(4)
IF(T.NE.0.0D0) TEMP(1) = E/T
31 ROOTS(1) = CMPLX(TEMP(1), 0.0D0, KIND=KIND(ROOTS))
ROOTS(2) = CMPLX(TEMP(2), 0.0D0, KIND=KIND(ROOTS))
ROOTS(3) = CMPLX(TEMP(3), 0.0D0, KIND=KIND(ROOTS))
ROOTS(4) = CMPLX(TEMP(4), 0.0D0, KIND=KIND(ROOTS))
RETURN
40 V1 = SQRT(ABS(XX(1)))
V2 = 0.0D0
GO TO 50
41 V1 = SQRT(ABS(XX(1)))
V2 = SQRT(ABS(XX(2)))
IF(Q < 0.0D0) U = -U
50 X = -U - B
Y = V1 - V2
ROOTS(1) = CMPLX(X, Y, KIND=KIND(ROOTS))
ROOTS(2) = CMPLX(X,-Y, KIND=KIND(ROOTS))
X = U - B
Y = V1 + V2
ROOTS(3) = CMPLX(X, Y, KIND=KIND(ROOTS))
ROOTS(4) = CMPLX(X,-Y, KIND=KIND(ROOTS))
RETURN
*----
* The resolvent cubic has complex roots.
*----
60 T = DBLE(ROOTS(1))
X = 0.0D0
IF(T < 0.0D0) THEN
GO TO 61
ELSE IF(T.EQ.0.0D0) THEN
GO TO 70
ELSE
GO TO 62
ENDIF
61 H = ABS(DBLE(ROOTS(2))) + ABS(AIMAG(ROOTS(2)))
IF(ABS(T).LE.H) GO TO 70
GO TO 80
62 X = SQRT(T)
IF(Q.GT.0.0D0) X = -X
70 W = SQRT(ROOTS(2))
U = 2.0D0*DBLE(W)
V = 2.0D0*ABS(AIMAG(W))
T = X - B
XX(1) = T + U
XX(2) = T - U
IF(ABS(XX(1)).LE.ABS(XX(2))) GO TO 71
T = XX(1)
XX(1) = XX(2)
XX(2) = T
71 U = -X - B
H = U*U + V*V
IF(XX(1)*XX(1) < 0.01D0*MIN(XX(2)*XX(2),H)) XX(1) = E/(XX(2)*H)
ROOTS(1) = CMPLX(XX(1), 0.0D0, KIND=KIND(ROOTS))
ROOTS(2) = CMPLX(XX(2), 0.0D0, KIND=KIND(ROOTS))
ROOTS(3) = CMPLX(U, V, KIND=KIND(ROOTS))
ROOTS(4) = CMPLX(U,-V, KIND=KIND(ROOTS))
RETURN
80 V = SQRT(ABS(T))
ROOTS(1) = CMPLX(-B, V, KIND=KIND(ROOTS))
ROOTS(2) = CMPLX(-B,-V, KIND=KIND(ROOTS))
ROOTS(3) = ROOTS(1)
ROOTS(4) = ROOTS(2)
RETURN
END
|
