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88 lines
2.6 KiB
C
88 lines
2.6 KiB
C
/* -*- Mode: C; comment-column: 22; fill-column: 79; -*- */
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#ifdef __cplusplus
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namespace Grid {
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namespace Approx {
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#endif
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#define HVERSION Header Time-stamp: <14-OCT-2004 09:26:51.00 adk@MISSCONTRARY>
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#ifndef ZOLOTAREV_INTERNAL
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#ifndef PRECISION
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#define PRECISION double
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#endif
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#define ZPRECISION PRECISION
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#define ZOLOTAREV_DATA zolotarev_data
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#endif
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/* This struct contains the coefficients which parameterise an optimal rational
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* approximation to the signum function.
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*
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* The parameterisations used are:
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*
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* Factored form for type 0 (R0(0) = 0)
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*
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* R0(x) = A * x * prod(x^2 - a[j], j = 0 .. dn-1) / prod(x^2 - ap[j], j = 0
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* .. dd-1),
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*
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* where deg_num = 2*dn + 1 and deg_denom = 2*dd.
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*
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* Factored form for type 1 (R1(0) = infinity)
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*
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* R1(x) = (A / x) * prod(x^2 - a[j], j = 0 .. dn-1) / prod(x^2 - ap[j], j = 0
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* .. dd-1),
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*
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* where deg_num = 2*dn and deg_denom = 2*dd + 1.
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*
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* Partial fraction form
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*
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* R(x) = alpha[da] * x + sum(alpha[j] * x / (x^2 - ap[j]), j = 0 .. da-1)
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*
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* where da = dd for type 0 and da = dd + 1 with ap[dd] = 0 for type 1.
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*
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* Continued fraction form
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*
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* R(x) = beta[db-1] * x + 1 / (beta[db-2] * x + 1 / (beta[db-3] * x + ...))
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*
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* with the final coefficient being beta[0], with d' = 2 * dd + 1 for type 0
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* and db = 2 * dd + 2 for type 1.
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*
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* Cayley form (Chiu's domain wall formulation)
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*
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* R(x) = (1 - T(x)) / (1 + T(x))
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*
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* where T(x) = prod((x - gamma[j]) / (x + gamma[j]), j = 0 .. n-1)
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*/
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typedef struct {
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ZPRECISION *a, /* zeros of numerator, a[0 .. dn-1] */
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*ap, /* poles (zeros of denominator), ap[0 .. dd-1] */
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A, /* overall factor */
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*alpha, /* coefficients of partial fraction, alpha[0 .. da-1] */
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*beta, /* coefficients of continued fraction, beta[0 .. db-1] */
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*gamma, /* zeros of numerator of T in Cayley form */
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Delta, /* maximum error, |R(x) - sgn(x)| <= Delta */
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epsilon; /* minimum x value, epsilon < |x| < 1 */
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int n, /* approximation degree */
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type, /* 0: R(0) = 0, 1: R(0) = infinity */
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dn, dd, da, db, /* number of elements of a, ap, alpha, and beta */
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deg_num, /* degree of numerator = deg_denom +/- 1 */
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deg_denom; /* degree of denominator */
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} ZOLOTAREV_DATA;
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#ifndef ZOLOTAREV_INTERNAL
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/* zolotarev(epsilon, n, type) returns a pointer to an initialised
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* zolotarev_data structure. The arguments must satisfy the constraints that
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* epsilon > 0, n > 0, and type = 0 or 1. */
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ZOLOTAREV_DATA* higham(PRECISION epsilon, int n) ;
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ZOLOTAREV_DATA* zolotarev(PRECISION epsilon, int n, int type);
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void zolotarev_free(zolotarev_data *zdata);
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#endif
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#ifdef __cplusplus
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}}
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#endif
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