void HermiteReduce(RatFun *g, RatFun *h, RatFun AD, char var, int trace)
{
Coefficient A = {special}, D = {special}, Dd = {special}, Dm = {special};
Coefficient Ds = {special}, Dmd = {special}, Dm2 = {special};
Coefficient Dms = {special}, B = {polynomial}, Bd = {special};
Coefficient C = {polynomial}, mDsDmd = {special};
RatFun old_g = *g, old_h = *h, t = {{special}, {special}};
g->num.type = rational;
g->num.u.rat = make_bigrat3(0);
g->den.type = rational;
g->den.u.rat = make_bigrat3(1);
init_ratfun(h);
B.u.poly = make_zero_poly('x');
C.u.poly = make_zero_poly('x');
copy_coefficient(&A, AD.num);
/* PRINTC(A); */
copy_coefficient(&D, AD.den);
/* PRINTC(D); */
coef_differentiate(&Dd, D, var);
/* PRINTC(Dd); */
coef_gcd(&Dm, D, Dd);
/* PRINTC(Dm); */
exact_div_coefficients(&Ds, D, Dm);
/* PRINTC(Ds); */
while (coef_deg(Dm, var) > 0) {
coef_differentiate(&Dmd, Dm, var);
/* PRINTC(Dmd); */
coef_gcd(&Dm2, Dm, Dmd);
/* PRINTC(Dm2); */
exact_div_coefficients(&Dms, Dm, Dm2);
/* PRINTC(Dms); */
mul_coefficients(&mDsDmd, Ds, Dmd);
exact_div_coefficients(&mDsDmd, mDsDmd, Dm);
negate_coefficient(&mDsDmd);
/* PRINTC(mDsDmd); */
/* PRINTC(Dms); */
/* PRINTC(A); */
if (trace) {
PRINT("Solving diophantine equation:\t");
printf("S * ");
print_coefficient(mDsDmd);
printf(" + T * ");
print_coefficient(Dms);
printf(" = ");
print_coefficient(A);
printf("\n\n");
}
SolveDiophantineEquation(&B, &C, mDsDmd, Dms, A);
if (trace) {
PRINT("Solution:\t\t\t");
printf("S = ");
print_coefficient(B);
printf(",\tT = ");
print_coefficient(C);
printf("\n");
WAIT;
}
/* PRINTC(B); */
/* PRINTC(C); */
/* PRINTC(Ds); */
/* PRINTC(Dms); */
exact_div_coefficients(&A, Ds, Dms);
/* PRINTC(A); */
/* PRINTC(B); */
coef_differentiate(&Bd, B, var);
/* PRINTC(Bd); */
mul_coefficients(&A, A, Bd);
sub_coefficients(&A, C, A);
/* PRINTC(A); */
t.num = B;
t.den = Dm;
/* PRINTR(*g); */
/* PRINTR(t); */
add_ratfuns(g, *g, t);
/* PRINTR(*g); */
copy_coefficient(&Dm, Dm2);
/* PRINTC(Dm); */
/* printf("-----\n"); */
}
h->num = A;
h->den = Ds;
canonicalise_ratfun(h);
free_ratfun(&old_g);
free_ratfun(&old_h);
}
void extract_polys(node_type **root)
{
node_type *r = *root;
Coefficient tq = {special}, tr = {special};
switch (r->type) {
case op2_type:
extract_polys(&(r->u.op2.operand1));
extract_polys(&(r->u.op2.operand2));
if (r->u.op2.operand1->type != coef_type
|| r->u.op2.operand2->type != coef_type) {
/* nothing to do here */
return;
}
switch (r->u.op2.operator) {
case '^':
if (r->u.op2.operand2->u.coef.type != rational
|| real_length(r->u.op2.operand2->u.coef.u.rat.num) != 1) {
printf("Error! "
"Indices must be single precision integers!\n");
exit(1);
}
if (coef_neg(r->u.op2.operand2->u.coef)) {
/* this is a rational function */
break;
}
coef_power(&r->u.op2.operand1->u.coef,
r->u.op2.operand1->u.coef,
*(r->u.op2.operand2->u.coef.u.rat.num+1));
*root = r->u.op2.operand1;
r->u.op2.operand1 = NULL;
free_tree(r);
break;
case '+':
add_coefficients(&r->u.op2.operand1->u.coef,
r->u.op2.operand1->u.coef,
r->u.op2.operand2->u.coef);
*root = r->u.op2.operand1;
r->u.op2.operand1 = NULL;
free_tree(r);
break;
case '-':
sub_coefficients(&r->u.op2.operand1->u.coef,
r->u.op2.operand1->u.coef,
r->u.op2.operand2->u.coef);
*root = r->u.op2.operand1;
r->u.op2.operand1 = NULL;
free_tree(r);
break;
case '*':
mul_coefficients(&r->u.op2.operand1->u.coef,
r->u.op2.operand1->u.coef,
r->u.op2.operand2->u.coef);
*root = r->u.op2.operand1;
r->u.op2.operand1 = NULL;
free_tree(r);
break;
case '/':
/* see if division will be exact */
polydiv_coefficients(&tq, &tr,
r->u.op2.operand1->u.coef,
r->u.op2.operand2->u.coef);
if (!coef_zero(tr)) {
/* division is not exact so this is a ratfun */
/* PRINTC(tq); */
/* PRINTC(tr); */
free_coefficient(&tq);
free_coefficient(&tr);
*root = add_ratfun(r->u.op2.operand1->u.coef,
r->u.op2.operand2->u.coef);
r->u.op2.operand1->u.coef.type = special;
r->u.op2.operand2->u.coef.type = special;
free_tree(r);
break;
}
/* division is exact */
copy_coefficient(&r->u.op2.operand1->u.coef, tq);
*root = r->u.op2.operand1;
r->u.op2.operand1 = NULL;
free_tree(r);
free_coefficient(&tq);
free_coefficient(&tr);
break;
}
break;
case op1_type:
extract_polys(&r->u.op1.operand);
if (r->u.op1.operand->type != coef_type) {
/* nothing to do here */
return;
}
switch (r->u.op1.operator) {
case UMINUS:
negate_coefficient(&r->u.op1.operand->u.coef);
*root = r->u.op1.operand;
r->u.op1.operand = NULL;
free_tree(r);
break;
}
break;
case coef_type:
case ratfun_type:
break;
}
}
void IntegrateRationalFunction(node_type *root, char var, char newvar,
int trace)
{
RatFun A, h, content;
Coefficient R = {special}, rat_part = {rational}, solution = {special};
Coefficient *Qit = NULL, *Sit = NULL;
Coefficient Qit2 = {special}, Sit2 = {special};
Integral integral;
unsigned i;
init_ratfun(&A);
init_ratfun(&h);
init_ratfun(&content);
init_bigrat(&rat_part.u.rat);
init_integral(&integral);
if (root->type != ratfun_type) {
printf("Error! IntegrateRationalFunction"
"requires a rational function.\n");
return;
}
copy_ratfun(&integral.integrand, root->u.ratfun);
integral.var = var;
integral.newvar = newvar;
if (ratfun_zero(integral.integrand)) {
goto print;
}
/* make numerator and denominator primitive */
coef_content(&content.num, root->u.ratfun.num, var);
exact_div_coefficients(&root->u.ratfun.num,
root->u.ratfun.num,
content.num);
coef_content(&content.den, root->u.ratfun.den, var);
exact_div_coefficients(&root->u.ratfun.den,
root->u.ratfun.den,
content.den);
if (trace) {
printf("Computing rational part of integral using "
"Hermite reduction...\n");
WAIT;
}
HermiteReduce(&integral.rational_part,
&h,
root->u.ratfun,
var,
trace);
/* put contents back on */
/* PRINTR(content); */
mul_ratfuns(&integral.rational_part, integral.rational_part, content);
mul_ratfuns(&h, h, content);
/* PRINTR(h); */
if (trace) {
printf("Found rational part:\t\t");
print_ratfun(integral.rational_part);
printf("\n");
printf("\nHermite reduction leaves:\t");
print_ratfun(h);
printf("\n");
WAIT;
printf("Remove polynomial part...\n");
WAIT;
}
polydiv_coefficients(&integral.poly_part, &R, h.num, h.den);
if (trace) {
printf("Polynomial part:\t\t");
print_coefficient(integral.poly_part);
printf("\n");
}
coef_integrate(&integral.poly_part, integral.poly_part, var);
if (trace) {
printf("Integrate it:\t\t\t");
print_coefficient(integral.poly_part);
printf("\n");
WAIT;
printf("Left over:\t\t\t");
print_coefficient(R);
printf("/");
print_coefficient(h.den);
printf("\n");
}
if (!coef_zero(R) && coef_deg(h.den, var) > coef_deg(R, var)) {
if (trace) {
printf("\nComputing logarithmic part using "
"Lazard-Rioboo-Trager algorithm...\n");
WAIT;
}
IntRationalLogPart(&integral.Qi, &integral.Si, R, h.den, var, newvar,
trace);
/* make Qi and Si primitive */
for (i = 0; i < integral.Qi.size; ++i) {
Qit = ca_get2(&integral.Qi, i);
Sit = ca_get2(&integral.Si, i);
rat_part.u.rat = coef_rat_part(*Qit);
mul_coefficients(Qit, *Qit, rat_part);
free_bigrat(&rat_part.u.rat);
/* rat_part.u.rat = coef_rat_part(*Sit); */
/* mul_coefficients(Sit, *Sit, rat_part); */
/* free_bigrat(&rat_part.u.rat); */
coef_pp(Qit, *Qit, newvar);
/* coef_pp(Sit, *Sit, var); */
}
if (trace) {
printf("Found logarithmic part.\n");
printf("%d sum(s) over roots:\t\t", integral.Qi.size);
if (integral.Qi.size > 0) {
for (i = 0; i < integral.Qi.size; ++i) {
printf("sum(%c | ", integral.newvar);
print_coefficient(ca_get(&integral.Qi, i));
printf(" = 0) %c*ln(", integral.newvar);
print_coefficient(ca_get(&integral.Si, i));
printf(")");
if (i < integral.Qi.size-1) {
printf(" + ");
}
}
}
printf("\n");
WAIT;
printf("Solve the linear univariate Qis to get explicit sums...\n");
}
/* solve linear univariate Qis */
for (i = 0; i < integral.Qi.size; ++i) {
if (coef_deg(ca_get(&integral.Qi, i), newvar) != 1
|| !poly_univar(ca_get(&integral.Qi, i).u.poly)) {
continue;
}
solve_linear_poly(&solution, ca_get(&integral.Qi, i).u.poly);
subst_var_coef(ca_get2(&integral.Si, i), solution, newvar);
/* move to solved arrays */
ca_push_back(&integral.QiS, solution);
ca_push_back(&integral.SiS, ca_get(&integral.Si, i));
/* remove from normal arrays */
Qit2 = ca_remove(&integral.Qi, i);
Sit2 = ca_remove(&integral.Si, i);
free_coefficient(&Qit2);
free_coefficient(&Sit2);
--i;
}
/* make SiS primitive */
for (i = 0; i < integral.SiS.size; ++i) {
Sit = ca_get2(&integral.SiS, i);
rat_part.u.rat = coef_rat_part(*Sit);
mul_coefficients(Sit, *Sit, rat_part);
free_bigrat(&rat_part.u.rat);
coef_pp(Sit, *Sit, var);
}
if (trace) {
printf("Found %d explicit sum(s):\t", integral.QiS.size);
if (integral.QiS.size > 0) {
for (i = 0; i < integral.QiS.size; ++i) {
if (!coef_one(ca_get(&integral.QiS, i))) {
print_coefficient(ca_get(&integral.QiS, i));
printf("*ln(");
}
else {
printf("ln(");
}
print_coefficient(ca_get(&integral.SiS, i));
printf(")");
if (i < integral.QiS.size-1) {
printf(" + ");
}
}
}
printf("\n");
WAIT;
}
}
else if (coef_deg(h.den, var) == 0) {
/* this is actually just a poly over a constant,
* so integrate it trivially */
coef_integrate(&R, R, var);
/* move this to the numerator of h */
free_coefficient(&h.num);
h.num = R;
R.type = special;
/* add h to the rational part of integral */
add_ratfuns(&integral.rational_part, integral.rational_part, h);
}
else {
/* this shouldn't happen */
printf("Error! Invalid ratfun following Hermite reduction!\n");
PRINTR(h);
printf("\n");
PRINTC(R);
printf("\n");
}
print:
if (trace) {
printf("\nFinal answer:\n");
}
print_integral(integral);
printf("\nLaTeX format:\n");
print_integral_LaTeX(integral);
free_ratfun(&A);
free_ratfun(&h);
free_ratfun(&content);
free_coefficient(&R);
free_integral(&integral);
}
inline void poly_invntt(poly* r)
{
static const uint16_t omegas_inv_montgomery[PARAM_N/2] =
{
4075, 5315, 4324, 4916, 10120, 11767, 7210, 9027, 10316, 6715, 1278, 9945,
3514, 11248, 11271, 5925, 147, 8500, 7840, 6833, 5537, 4749, 4467, 7500,
11099, 9606, 6171, 8471, 8429, 5445, 11239, 7753, 9090, 12233, 5529, 5206,
10587, 1987, 11635, 3565, 5415, 8646, 6153, 6427, 7341, 6152, 10561, 400,
8410, 1922, 2033, 8291, 1359, 6854, 11035, 973, 8579, 6093, 6950, 5446,
11821, 8301, 11907, 316, 52, 3174, 10966, 9523, 6055, 8953, 11612, 6415,
2505, 5906, 10710, 11858, 8332, 9450, 10162, 151, 3482, 787, 5468, 1010,
4169, 9162, 5241, 9369, 7509, 8844, 7232, 4698, 192, 1321, 10240, 4912, 885,
6281, 10333, 7280, 8757, 11286, 58, 12048, 12147, 11184, 8812, 6608, 2844,
3438, 4212, 11314, 8687, 6068, 421, 8209, 3600, 3263, 7665, 6077, 7507, 5886,
3029, 6695, 4213, 504, 11684, 2302, 1962, 1594, 6328, 7183, 168, 2692, 8960,
4298, 5184, 11089, 6122, 9734, 10929, 3956, 5297, 6170, 3762, 9370, 4016,
4077, 6523, 652, 11994, 6099, 1146, 11341, 11964, 10885, 6299, 1159, 8240,
8561, 11177, 2078, 10331, 4322, 11367, 441, 4079, 11231, 3150, 1319, 8243,
709, 8049, 8719, 11454, 6224, 3054, 6803, 3123, 10542, 4433, 6370, 7032,
3834, 8633, 12225, 9830, 683, 1566, 5782, 9786, 9341, 12115, 723, 3009, 1693,
5735, 2655, 2738, 6421, 11942, 2925, 1975, 8532, 3315, 11863, 4754, 1858,
1583, 6347, 2500, 10800, 6374, 1483, 12240, 1263, 1815, 5383, 10777, 350,
6920, 10232, 4493, 9087, 8855, 8760, 9381, 218, 9928, 10446, 9259, 4115,
6147, 9842, 8326, 576, 10335, 10238, 10484, 9407, 6381, 11836, 8517, 418,
6860, 7515, 1293, 7552, 2767, 156, 8298, 8320, 10008, 5876, 5333, 10258,
10115, 4372, 2847, 7875, 8232, 9018, 8925, 1689, 8236, 2645, 5042, 9984,
7094, 9509, 1484, 7394, 3, 4437, 160, 3149, 113, 7370, 10123, 3915, 6998,
2704, 8653, 4938, 1426, 7635, 10512, 1663, 6957, 3510, 2370, 2865, 3978,
9320, 3247, 9603, 6882, 3186, 10659, 10163, 1153, 9405, 8241, 10040, 2178,
1544, 5559, 420, 8304, 4905, 476, 3531, 5191, 9153, 2399, 8889, 3000, 671,
243, 3016, 3763, 10849, 12262, 9223, 10657, 7205, 11272, 7404, 7575, 8146,
10752, 242, 2678, 3704, 11744, 5019, 3833, 3778, 11899, 773, 5101, 11222,
9888, 442, 2912, 5698, 11935, 4861, 7277, 9808, 11244, 2859, 3780, 11414,
4976, 10682, 7201, 8005, 11287, 5011, 6267, 2987, 2437, 3646, 2566, 10102,
9867, 6250, 5444, 2381, 11796, 8193, 4337, 11854, 1912, 1378, 404, 7644,
1065, 2143, 11121, 5277, 3248, 11082, 2548, 8058, 8907, 11934, 1759, 8582,
3694, 7110, 12144, 6747, 8652, 3459, 2731, 8357, 6378, 7399, 10861, 1696,
9863, 334, 7657, 6534, 11029, 4388, 11560, 3241, 10276, 9000, 9408, 3284,
10200, 7197, 6498, 544, 2468, 339, 11267, 9, 2842, 480, 5331, 7300, 1673,
4278, 4177, 8705, 9764, 1381, 7837, 2396, 8340, 8993, 4354, 130, 6915, 2837,
11462, 5767, 953, 8541, 9813, 118, 7222, 2197, 3006, 9545, 563, 9314, 2625,
11340, 4821, 2639, 7266, 5828, 6561, 7698, 3328, 6512, 1351, 7311, 6553,
8155, 1305, 722, 5146, 4043, 12288, 10810, 2545, 3621, 8747, 8785, 1646,
1212, 5860, 3195, 7203, 10963, 3201, 3014, 955, 11499, 9970, 11119, 3135,
3712, 7443, 9542, 7484, 8736, 9995, 11227, 1635, 9521, 1177, 8034, 140,
10436, 11563, 7678, 4320, 11289, 9198, 12208, 2963, 7393, 2366, 9238
};
static const uint16_t psis_inv_montgomery[PARAM_N] =
{
256, 10570, 1510, 7238, 1034, 7170, 6291, 7921, 11665, 3422, 4000, 2327,
2088, 5565, 795, 10647, 1521, 5484, 2539, 7385, 1055, 7173, 8047, 11683,
1669, 1994, 3796, 5809, 4341, 9398, 11876, 12230, 10525, 12037, 12253, 3506,
4012, 9351, 4847, 2448, 7372, 9831, 3160, 2207, 5582, 2553, 7387, 6322, 9681,
1383, 10731, 1533, 219, 5298, 4268, 7632, 6357, 9686, 8406, 4712, 9451,
10128, 4958, 5975, 11387, 8649, 11769, 6948, 11526, 12180, 1740, 10782, 6807,
2728, 7412, 4570, 4164, 4106, 11120, 12122, 8754, 11784, 3439, 5758, 11356,
6889, 9762, 11928, 1704, 1999, 10819, 12079, 12259, 7018, 11536, 1648, 1991,
2040, 2047, 2048, 10826, 12080, 8748, 8272, 8204, 1172, 1923, 7297, 2798,
7422, 6327, 4415, 7653, 6360, 11442, 12168, 7005, 8023, 9924, 8440, 8228,
2931, 7441, 1063, 3663, 5790, 9605, 10150, 1450, 8985, 11817, 10466, 10273,
12001, 3470, 7518, 1074, 1909, 7295, 9820, 4914, 702, 5367, 7789, 8135, 9940,
1420, 3714, 11064, 12114, 12264, 1752, 5517, 9566, 11900, 1700, 3754, 5803,
829, 1874, 7290, 2797, 10933, 5073, 7747, 8129, 6428, 6185, 11417, 1631, 233,
5300, 9535, 10140, 11982, 8734, 8270, 2937, 10953, 8587, 8249, 2934, 9197,
4825, 5956, 4362, 9401, 1343, 3703, 529, 10609, 12049, 6988, 6265, 895, 3639,
4031, 4087, 4095, 585, 10617, 8539, 4731, 4187, 9376, 3095, 9220, 10095,
10220, 1460, 10742, 12068, 1724, 5513, 11321, 6884, 2739, 5658, 6075, 4379,
11159, 10372, 8504, 4726, 9453, 3106, 7466, 11600, 10435, 8513, 9994, 8450,
9985, 3182, 10988, 8592, 2983, 9204, 4826, 2445, 5616, 6069, 867, 3635, 5786,
11360, 5134, 2489, 10889, 12089, 1727, 7269, 2794, 9177, 1311, 5454, 9557,
6632, 2703, 9164, 10087, 1441, 3717, 531, 3587, 2268, 324, 5313, 759, 1864,
5533, 2546, 7386, 9833, 8427, 4715, 11207, 1601, 7251, 4547, 11183, 12131,
1733, 10781, 10318, 1474, 10744, 5046, 4232, 11138, 10369, 6748, 964, 7160,
4534, 7670, 8118, 8182, 4680, 11202, 6867, 981, 8918, 1274, 182, 26, 7026,
8026, 11680, 12202, 10521, 1503, 7237, 4545, 5916, 9623, 8397, 11733, 10454,
3249, 9242, 6587, 941, 1890, 270, 10572, 6777, 9746, 6659, 6218, 6155, 6146,
878, 1881, 7291, 11575, 12187, 1741, 7271, 8061, 11685, 6936, 4502, 9421,
4857, 4205, 7623, 1089, 10689, 1527, 8996, 10063, 11971, 10488, 6765, 2722,
3900, 9335, 11867, 6962, 11528, 5158, 4248, 4118, 5855, 2592, 5637, 6072,
2623, 7397, 8079, 9932, 4930, 5971, 853, 3633, 519, 8852, 11798, 3441, 11025,
1575, 225, 8810, 11792, 12218, 3501, 9278, 3081, 9218, 4828, 7712, 8124,
11694, 12204, 3499, 4011, 573, 3593, 5780, 7848, 9899, 10192, 1456, 208,
7052, 2763, 7417, 11593, 10434, 12024, 8740, 11782, 10461, 3250, 5731, 7841,
9898, 1414, 202, 3540, 7528, 2831, 2160, 10842, 5060, 4234, 4116, 588, 84,
12, 7024, 2759, 9172, 6577, 11473, 1639, 9012, 3043, 7457, 6332, 11438, 1634,
1989, 9062, 11828, 8712, 11778, 12216, 10523, 6770, 9745, 10170, 4964, 9487,
6622, 946, 8913, 6540, 6201, 4397, 9406, 8366, 9973, 8447, 8229, 11709, 8695,
10020, 3187, 5722, 2573, 10901, 6824, 4486, 4152, 9371, 8361, 2950, 2177,
311, 1800, 9035, 8313, 11721, 3430, 490, 70, 10, 1757, 251, 3547, 7529,
11609, 3414, 7510, 4584, 4166, 9373, 1339, 5458, 7802, 11648, 1664, 7260,
9815, 10180, 6721, 9738, 10169, 8475, 8233, 9954, 1422, 8981, 1283, 5450,
11312, 1616, 3742, 11068, 10359, 4991, 713, 3613, 9294, 8350, 4704, 672, 96,
7036, 9783, 11931, 3460, 5761, 823, 10651, 12055, 10500, 1500, 5481, 783,
3623, 11051, 8601, 8251, 8201, 11705, 10450, 5004, 4226, 7626, 2845, 2162,
3820, 7568, 9859, 3164, 452, 10598, 1514, 5483, 6050, 6131, 4387, 7649, 8115,
6426, 918, 8909, 8295, 1185, 5436, 11310, 8638, 1234, 5443, 11311, 5127,
2488, 2111, 10835, 5059, 7745, 2862, 3920, 560, 80, 1767, 2008, 3798, 11076,
6849, 2734, 10924, 12094, 8750, 1250, 10712, 6797, 971, 7161, 1023, 8924,
4786, 7706, 4612, 4170, 7618, 6355, 4419, 5898, 11376, 10403, 10264, 6733,
4473, 639, 5358, 2521, 9138, 3061, 5704, 4326, 618, 5355, 765, 5376, 768,
7132, 4530, 9425, 3102, 9221, 6584, 11474, 10417, 10266, 12000, 6981, 6264,
4406, 2385, 7363, 4563, 4163, 7617, 9866, 3165, 9230, 11852, 10471, 5007,
5982, 11388, 5138, 734, 3616, 11050, 12112, 6997, 11533, 12181, 10518, 12036,
3475, 2252, 7344, 9827, 4915, 9480, 6621, 4457, 7659, 9872, 6677, 4465, 4149,
7615, 4599, 657, 3605, 515, 10607, 6782, 4480, 640, 1847, 3775, 5806, 2585,
5636, 9583, 1369, 10729, 8555, 10000, 11962, 5220, 7768, 8132, 8184, 9947,
1421, 203, 29, 8782, 11788, 1684, 10774, 10317, 4985, 9490, 8378, 4708,
11206, 5112, 5997, 7879, 11659, 12199, 8765, 10030, 4944, 5973, 6120, 6141,
6144, 7900, 11662, 1666, 238, 34, 3516, 5769, 9602, 8394, 9977, 6692, 956,
10670, 6791, 9748, 11926, 8726, 11780, 5194, 742, 106, 8793, 10034, 3189,
10989, 5081, 4237, 5872, 4350, 2377, 10873, 6820, 6241, 11425, 10410, 10265,
3222, 5727, 9596, 4882, 2453, 2106, 3812, 11078, 12116, 5242, 4260, 11142,
8614, 11764, 12214, 5256, 4262, 4120, 11122, 5100, 11262, 5120, 2487, 5622,
9581, 8391, 8221, 2930, 10952, 12098, 6995, 6266, 9673, 4893, 699, 3611,
4027, 5842, 11368, 1624, 232, 8811, 8281, 1183, 169, 8802, 3013, 2186, 5579,
797, 3625, 4029, 11109, 1587, 7249, 11569, 8675, 6506, 2685, 10917, 12093,
12261, 12285, 1755, 7273, 1039, 1904, 272, 3550, 9285, 3082, 5707, 6082,
4380, 7648, 11626, 5172, 4250, 9385, 8363, 8217, 4685, 5936, 848, 8899, 6538,
934, 1889, 3781, 9318, 10109, 10222, 6727, 961, 5404, 772, 5377, 9546, 8386,
1198, 8949, 3034, 2189, 7335, 4559, 5918, 2601, 10905, 5069, 9502, 3113,
7467, 8089, 11689, 5181, 9518, 8382, 2953, 3933, 4073, 4093, 7607, 8109,
2914, 5683, 4323, 11151, 1593, 10761, 6804, 972, 3650, 2277, 5592, 4310,
7638, 9869, 4921, 703, 1856, 9043, 4803, 9464, 1352, 8971, 11815, 5199, 7765,
6376, 4422, 7654, 2849, 407, 8836, 6529, 7955, 2892, 9191, 1313, 10721,
12065, 12257, 1751, 9028, 8312, 2943, 2176, 3822, 546, 78, 8789, 11789,
10462, 12028, 6985, 4509, 9422, 1346, 5459, 4291, 613, 10621, 6784, 9747,
3148, 7472, 2823, 5670, 810, 7138, 8042, 4660, 7688, 6365, 6176, 6149, 2634,
5643, 9584, 10147, 11983, 5223, 9524, 11894, 10477, 8519, 1217, 3685, 2282,
326, 10580, 3267, 7489, 4581, 2410, 5611, 11335, 6886, 8006, 8166, 11700,
3427, 11023, 8597, 10006, 3185, 455, 65, 5276, 7776, 4622, 5927, 7869, 9902,
11948, 5218, 2501, 5624, 2559, 10899, 1557, 1978, 10816, 10323, 8497, 4725,
675, 1852, 10798, 12076, 10503, 3256, 9243, 3076, 2195, 10847, 12083, 10504,
12034, 10497
};
bitrev_vector(r->coeffs);
ntt(r->coeffs, omegas_inv_montgomery);
mul_coefficients(r->coeffs, psis_inv_montgomery);
}
