1ZLATPS(1) LAPACK auxiliary routine (version 3.2) ZLATPS(1)
2
6 ZLATPS - solves one of the triangular systems A * x = s*b, A**T * x =
7 s*b, or A**H * x = s*b,
8
10 SUBROUTINE ZLATPS( UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM,
11 INFO )
12 CHARACTER DIAG, NORMIN, TRANS, UPLO
14 INTEGER INFO, N
16 DOUBLE PRECISION SCALE
18 DOUBLE PRECISION CNORM( * )
20 COMPLEX*16 AP( * ), X( * )
22
24 ZLATPS solves one of the triangular systems with scaling to prevent
25 overflow, where A is an upper or lower triangular matrix stored in
26 packed form. Here A**T denotes the transpose of A, A**H denotes the
27 conjugate transpose of A, x and b are n-element vectors, and s is a
28 scaling factor, usually less than or equal to 1, chosen so that the
29 components of x will be less than the overflow threshold. If the
30 unscaled problem will not cause overflow, the Level 2 BLAS routine
31 ZTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
32 then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
33
35 UPLO (input) CHARACTER*1
36 Specifies whether the matrix A is upper or lower triangular. =
37 'U': Upper triangular
38 = 'L': Lower triangular
39 TRANS (input) CHARACTER*1
41 Specifies the operation applied to A. = 'N': Solve A * x =
42 s*b (No transpose)
43 = 'T': Solve A**T * x = s*b (Transpose)
44 = 'C': Solve A**H * x = s*b (Conjugate transpose)
45 DIAG (input) CHARACTER*1
47 Specifies whether or not the matrix A is unit triangular. =
48 'N': Non-unit triangular
49 = 'U': Unit triangular
50 NORMIN (input) CHARACTER*1
52 Specifies whether CNORM has been set or not. = 'Y': CNORM
53 contains the column norms on entry
54 = 'N': CNORM is not set on entry. On exit, the norms will be
55 computed and stored in CNORM.
56 N (input) INTEGER
58 The order of the matrix A. N >= 0.
59 AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
61 The upper or lower triangular matrix A, packed columnwise in a
62 linear array. The j-th column of A is stored in the array AP
63 as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for
64 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for
65 j<=i<=n.
66 X (input/output) COMPLEX*16 array, dimension (N)
68 On entry, the right hand side b of the triangular system. On
69 exit, X is overwritten by the solution vector x.
70 SCALE (output) DOUBLE PRECISION
72 The scaling factor s for the triangular system A * x = s*b,
73 A**T * x = s*b, or A**H * x = s*b. If SCALE = 0, the matrix
74 A is singular or badly scaled, and the vector x is an exact or
75 approximate solution to A*x = 0.
76 CNORM (input or output) DOUBLE PRECISION array, dimension (N)
78 If NORMIN = 'Y', CNORM is an input argument and CNORM(j) con‐
79 tains the norm of the off-diagonal part of the j-th column of
80 A. If TRANS = 'N', CNORM(j) must be greater than or equal to
81 the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be
82 greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is
83 an output argument and CNORM(j) returns the 1-norm of the off‐
84 diagonal part of the j-th column of A.
85 INFO (output) INTEGER
87 = 0: successful exit
88 < 0: if INFO = -k, the k-th argument had an illegal value
89
91 A rough bound on x is computed; if that is less than overflow, ZTPSV is
92 called, otherwise, specific code is used which checks for possible
93 overflow or divide-by-zero at every operation.
94 A columnwise scheme is used for solving A*x = b. The basic algorithm
95 if A is lower triangular is
96 x[1:n] := b[1:n]
97 for j = 1, ..., n
98 x(j) := x(j) / A(j,j)
99 x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
100 end
101 Define bounds on the components of x after j iterations of the loop:
102 M(j) = bound on x[1:j]
103 G(j) = bound on x[j+1:n]
104 Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
105 Then for iteration j+1 we have
106 M(j+1) <= G(j) / | A(j+1,j+1) |
107 G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
108 <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
109 where CNORM(j+1) is greater than or equal to the infinity-norm of col‐
110 umn j+1 of A, not counting the diagonal. Hence
111 G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
112 1<=i<=j
113 and
114 |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
115 1<=i< j
116 Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the
117 reciprocal of the largest M(j), j=1,..,n, is larger than
118 max(underflow, 1/overflow).
119 The bound on x(j) is also used to determine when a step in the column‐
120 wise method can be performed without fear of overflow. If the computed
121 bound is greater than a large constant, x is scaled to prevent over‐
122 flow, but if the bound overflows, x is set to 0, x(j) to 1, and scale
123 to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a
124 row-wise scheme is used to solve A**T *x = b or A**H *x = b. The
125 basic algorithm for A upper triangular is
126 for j = 1, ..., n
127 x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
128 end
129 We simultaneously compute two bounds
130 G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
131 M(j) = bound on x(i), 1<=i<=j
132 The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add
133 the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the
134 bound on x(j) is
135 M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
136 <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
137 1<=i<=j
138 and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater than
139 max(underflow, 1/overflow).
140 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 ZLATPS(1)