// Compute the square root of n using Newton's Method. We start with
// an initial estimate for sqrt(n), and then iterate
// x_{i+1} = 1/2 * ( x_i + (n / x_i) )
// Result is returned in x
func (e *Pslq) Sqrt(n, x *big.Float) {
if n == x {
panic("need distinct input and output")
}
if n.Sign() == 0 {
x.Set(n)
return
} else if n.Sign() < 0 {
panic("Sqrt of negative number")
}
prec := n.Prec()
// Use the floating point square root as initial estimate
nFloat64, _ := n.Float64()
x.SetPrec(prec).SetFloat64(math.Sqrt(nFloat64))
// We use t as a temporary variable. There's no need to set its precision
// since big.Float values with unset (== 0) precision automatically assume
// the largest precision of the arguments when used as the result (receiver)
// of a big.Float operation.
var t big.Float
// Iterate.
for {
t.Quo(n, x) // t = n / x_i
t.Add(x, &t) // t = x_i + (n / x_i)
t.Mul(&e.half, &t) // x_{i+1} = 0.5 * t
if x.Cmp(&t) == 0 {
// Exit loop if no change to result
break
}
x.Set(&t)
}
}
func makeFloat(x *big.Float) Value {
// convert -0
if x.Sign() == 0 {
return floatVal0
}
return floatVal{x}
}
// BigCommaf produces a string form of the given big.Float in base 10
// with commas after every three orders of magnitude.
func BigCommaf(v *big.Float) string {
buf := &bytes.Buffer{}
if v.Sign() < 0 {
buf.Write([]byte{'-'})
v.Abs(v)
}
comma := []byte{','}
parts := strings.Split(v.Text('f', -1), ".")
pos := 0
if len(parts[0])%3 != 0 {
pos += len(parts[0]) % 3
buf.WriteString(parts[0][:pos])
buf.Write(comma)
}
for ; pos < len(parts[0]); pos += 3 {
buf.WriteString(parts[0][pos : pos+3])
buf.Write(comma)
}
buf.Truncate(buf.Len() - 1)
if len(parts) > 1 {
buf.Write([]byte{'.'})
buf.WriteString(parts[1])
}
return buf.String()
}
// floatSqrt computes the square root of x using Newton's method.
// TODO: Use a better algorithm such as the one from math/sqrt.go.
func floatSqrt(c Context, x *big.Float) *big.Float {
switch x.Sign() {
case -1:
Errorf("square root of negative number")
case 0:
return newFloat(c)
}
// Each iteration computes
// z = z - (z²-x)/2z
// z holds the result so far. A good starting point is to halve the exponent.
// Experiments show we converge in only a handful of iterations.
z := newFloat(c)
exp := x.MantExp(z)
z.SetMantExp(z, exp/2)
// Intermediates, allocated once.
zSquared := newFloat(c)
num := newFloat(c)
den := newFloat(c)
for loop := newLoop(c.Config(), "sqrt", x, 1); ; {
zSquared.Mul(z, z)
num.Sub(zSquared, x)
den.Mul(floatTwo, z)
num.Quo(num, den)
z.Sub(z, num)
if loop.done(z) {
break
}
}
return z
}
// floatLog computes natural log(x) using the Maclaurin series for log(1-x).
func floatLog(x *big.Float) *big.Float {
if x.Sign() <= 0 {
Errorf("log of non-positive value")
}
// The series wants x < 1, and log 1/x == -log x, so exploit that.
invert := false
if x.Cmp(floatOne) > 0 {
invert = true
xx := newF()
xx.Quo(floatOne, x)
x = xx
}
// x = mantissa * 2**exp, and 0.5 <= mantissa < 1.
// So log(x) is log(mantissa)+exp*log(2), and 1-x will be
// between 0 and 0.5, so the series for 1-x will converge well.
// (The series converges slowly in general.)
mantissa := newF()
exp2 := x.MantExp(mantissa)
exp := newF().SetInt64(int64(exp2))
exp.Mul(exp, floatLog2)
if invert {
exp.Neg(exp)
}
// y = 1-x (whereupon x = 1-y and we use that in the series).
y := newF().SetInt64(1)
y.Sub(y, mantissa)
// The Maclaurin series for log(1-y) == log(x) is: -y - y²/2 - y³/3 ...
yN := newF().Set(y)
term := newF()
n := newF().Set(floatOne)
z := newF()
// This is the slowest-converging series, so we add a factor of ten to the cutoff.
// Only necessary when FloatPrec is at or beyond constPrecisionInBits.
loop := newLoop("log", y, 40)
for {
term.Set(yN)
term.Quo(term, n)
z.Sub(z, term)
if loop.terminate(z) {
break
}
// Advance y**index (multiply by y).
yN.Mul(yN, y)
n.Add(n, floatOne)
}
if invert {
z.Neg(z)
}
z.Add(z, exp)
return z
}
// NearestInt set res to the nearest integer to x
func (e *Pslq) NearestInt(x *big.Float, res *big.Int) {
prec := x.Prec()
var tmp big.Float
tmp.SetPrec(prec)
if x.Sign() >= 0 {
tmp.Add(x, &e.half)
} else {
tmp.Sub(x, &e.half)
}
tmp.Int(res)
}
// floatSin computes sin(x) using argument reduction and a Taylor series.
func floatSin(c Context, x *big.Float) *big.Float {
negate := false
if x.Sign() < 0 {
x.Neg(x)
negate = true
}
twoPiReduce(c, x)
// sin(x) = x - x³/3! + x⁵/5! - ...
// First term to compute in loop will be -x³/3!
factorial := newFloat(c).SetInt64(6)
result := sincos("sin", c, 3, x, newFloat(c).Set(x), 3, factorial)
if negate {
result.Neg(result)
}
return result
}
// floatSqrt computes the square root of x using Newton's method.
// TODO: Use a better algorithm such as the one from math/sqrt.go.
func floatSqrt(x *big.Float) *big.Float {
switch x.Sign() {
case -1:
Errorf("square root of negative number")
case 0:
return newF()
}
// Each iteration computes
// z = z - (z²-x)/2z
// delta holds the difference between the result
// this iteration and the previous. The loop stops
// when it hits zero.
// z holds the result so far. A good starting point is to halve the exponent.
// Experiments show we converge in only a handful of iterations.
z := newF()
exp := x.MantExp(z)
z.SetMantExp(z, exp/2)
// Intermediates, allocated once.
zSquared := newF()
num := newF()
den := newF()
loop := newLoop("sqrt", x, 1)
for {
zSquared.Mul(z, z)
num.Sub(zSquared, x)
den.Mul(floatTwo, z)
num.Quo(num, den)
z.Sub(z, num)
if loop.terminate(z) {
break
}
}
return z
}
// Sqrt returns a big.Float representation of the square root of
// z. Precision is the same as the one of the argument. The function
// panics if z is negative, returns ±0 when z = ±0, and +Inf when z =
// +Inf.
func Sqrt(z *big.Float) *big.Float {
// panic on negative z
if z.Sign() == -1 {
panic("Sqrt: argument is negative")
}
// √±0 = ±0
if z.Sign() == 0 {
return big.NewFloat(float64(z.Sign()))
}
// √+Inf = +Inf
if z.IsInf() {
return big.NewFloat(math.Inf(+1))
}
// Compute √(a·2**b) as
// √(a)·2**b/2 if b is even
// √(2a)·2**b/2 if b > 0 is odd
// √(0.5a)·2**b/2 if b < 0 is odd
//
// The difference in the odd exponent case is due to the fact that
// exp/2 is rounded in different directions when exp is negative.
mant := new(big.Float)
exp := z.MantExp(mant)
switch exp % 2 {
case 1:
mant.Mul(big.NewFloat(2), mant)
case -1:
mant.Mul(big.NewFloat(0.5), mant)
}
// Solving x² - z = 0 directly requires a Quo call, but it's
// faster for small precisions.
//
// Solving 1/x² - z = 0 avoids the Quo call and is much faster for
// high precisions.
//
// Use sqrtDirect for prec <= 128 and sqrtInverse for prec > 128.
var x *big.Float
if z.Prec() <= 128 {
x = sqrtDirect(mant)
} else {
x = sqrtInverse(mant)
}
// re-attach the exponent and return
return x.SetMantExp(x, exp/2)
}
// Exp returns a big.Float representation of exp(z). Precision is
// the same as the one of the argument. The function returns +Inf
// when z = +Inf, and 0 when z = -Inf.
func Exp(z *big.Float) *big.Float {
// exp(0) == 1
if z.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Exp(+Inf) = +Inf
if z.IsInf() && z.Sign() > 0 {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
// Exp(-Inf) = 0
if z.IsInf() && z.Sign() < 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
guess := new(big.Float)
// try to get initial estimate using IEEE-754 math
zf, _ := z.Float64()
if zfs := math.Exp(zf); zfs == math.Inf(+1) || zfs == 0 {
// too big or too small for IEEE-754 math,
// perform argument reduction using
// e^{2z} = (e^z)²
halfZ := new(big.Float).Mul(z, big.NewFloat(0.5))
halfExp := Exp(halfZ.SetPrec(z.Prec() + 64))
return new(big.Float).Mul(halfExp, halfExp).SetPrec(z.Prec())
} else {
// we got a nice IEEE-754 estimate
guess.SetFloat64(zfs)
}
// f(t)/f'(t) = t*(log(t) - z)
f := func(t *big.Float) *big.Float {
x := new(big.Float)
x.Sub(Log(t), z)
return x.Mul(x, t)
}
x := newton(f, guess, z.Prec())
return x
}
// Pow returns a big.Float representation of z**w. Precision is the same as the one
// of the first argument. The function panics when z is negative.
func Pow(z *big.Float, w *big.Float) *big.Float {
if z.Sign() < 0 {
panic("Pow: negative base")
}
// Pow(z, 0) = 1.0
if w.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Pow(z, 1) = z
// Pow(+Inf, n) = +Inf
if w.Cmp(big.NewFloat(1)) == 0 || z.IsInf() {
return new(big.Float).Copy(z)
}
// Pow(z, -w) = 1 / Pow(z, w)
if w.Sign() < 0 {
x := new(big.Float)
zExt := new(big.Float).Copy(z).SetPrec(z.Prec() + 64)
wNeg := new(big.Float).Neg(w)
return x.Quo(big.NewFloat(1), Pow(zExt, wNeg)).SetPrec(z.Prec())
}
// w integer fast path
if w.IsInt() {
wi, _ := w.Int64()
return powInt(z, int(wi))
}
// compute w**z as exp(z log(w))
x := new(big.Float).SetPrec(z.Prec() + 64)
logZ := Log(new(big.Float).Copy(z).SetPrec(z.Prec() + 64))
x.Mul(w, logZ)
x = Exp(x)
return x.SetPrec(z.Prec())
}
// Log returns a big.Float representation of the natural logarithm of
// z. Precision is the same as the one of the argument. The function
// panics if z is negative, returns -Inf when z = 0, and +Inf when z =
// +Inf
func Log(z *big.Float) *big.Float {
// panic on negative z
if z.Sign() == -1 {
panic("Log: argument is negative")
}
// Log(0) = -Inf
if z.Sign() == 0 {
return big.NewFloat(math.Inf(-1)).SetPrec(z.Prec())
}
prec := z.Prec() + 64 // guard digits
one := big.NewFloat(1).SetPrec(prec)
two := big.NewFloat(2).SetPrec(prec)
four := big.NewFloat(4).SetPrec(prec)
// Log(1) = 0
if z.Cmp(one) == 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
// Log(+Inf) = +Inf
if z.IsInf() {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
x := new(big.Float).SetPrec(prec)
// if 0 < z < 1 we compute log(z) as -log(1/z)
var neg bool
if z.Cmp(one) < 0 {
x.Quo(one, z)
neg = true
} else {
x.Set(z)
}
// We scale up x until x >= 2**(prec/2), and then we'll be allowed
// to use the AGM formula for Log(x).
//
// Double x until the condition is met, and keep track of the
// number of doubling we did (needed to scale back later).
lim := new(big.Float)
lim.SetMantExp(two, int(prec/2))
k := 0
for x.Cmp(lim) < 0 {
x.Mul(x, x)
k++
}
// Compute the natural log of x using the fact that
// log(x) = π / (2 * AGM(1, 4/x))
// if
// x >= 2**(prec/2),
// where prec is the desired precision (in bits)
pi := pi(prec)
agm := agm(one, x.Quo(four, x)) // agm = AGM(1, 4/x)
x.Quo(pi, x.Mul(two, agm)) // reuse x, we don't need it
if neg {
x.Neg(x)
}
// scale the result back multiplying by 2**-k
// reuse lim to reduce allocations.
x.Mul(x, lim.SetMantExp(one, -k))
return x.SetPrec(z.Prec())
}
// floatAtan computes atan(x) using a Taylor series. There are two series,
// one for |x| < 1 and one for larger values.
func floatAtan(c Context, x *big.Float) *big.Float {
// atan(-x) == -atan(x). Do this up top to simplify the Euler crossover calculation.
if x.Sign() < 0 {
z := newFloat(c).Set(x)
z = floatAtan(c, z.Neg(z))
return z.Neg(z)
}
// The series converge very slowly near 1. atan 1.00001 takes over a million
// iterations at the default precision. But there is hope, an Euler identity:
// atan(x) = atan(y) + atan((x-y)/(1+xy))
// Note that y is a free variable. If x is near 1, we can use this formula
// to push the computation to values that converge faster. Because
// tan(π/8) = √2 - 1, or equivalently atan(√2 - 1) == π/8
// we choose y = √2 - 1 and then we only need to calculate one atan:
// atan(x) = π/8 + atan((x-y)/(1+xy))
// Where do we cross over? This version converges significantly faster
// even at 0.5, but we must be careful that (x-y)/(1+xy) never approaches 1.
// At x = 0.5, (x-y)/(1+xy) is 0.07; at x=1 it is 0.414214; at x=1.5 it is
// 0.66, which is as big as we dare go. With 256 bits of precision and a
// crossover at 0.5, here are the number of iterations done by
// atan .1*iota 20
// 0.1 39, 0.2 55, 0.3 73, 0.4 96, 0.5 126, 0.6 47, 0.7 59, 0.8 71, 0.9 85, 1.0 99, 1.1 116, 1.2 38, 1.3 44, 1.4 50, 1.5 213, 1.6 183, 1.7 163, 1.8 147, 1.9 135, 2.0 125
tmp := newFloat(c).Set(floatOne)
tmp.Sub(tmp, x)
tmp.Abs(tmp)
if tmp.Cmp(newFloat(c).SetFloat64(0.5)) < 0 {
z := newFloat(c).Set(floatPi)
z.Quo(z, newFloat(c).SetInt64(8))
y := floatSqrt(c, floatTwo)
y.Sub(y, floatOne)
num := newFloat(c).Set(x)
num.Sub(num, y)
den := newFloat(c).Set(x)
den = den.Mul(den, y)
den = den.Add(den, floatOne)
z = z.Add(z, floatAtan(c, num.Quo(num, den)))
return z
}
if x.Cmp(floatOne) > 0 {
return floatAtanLarge(c, x)
}
// This is the series for small values |x| < 1.
// asin(x) = x - x³/3 + x⁵/5 - x⁷/7 + ...
// First term to compute in loop will be x
n := newFloat(c)
term := newFloat(c)
xN := newFloat(c).Set(x)
xSquared := newFloat(c).Set(x)
xSquared.Mul(x, x)
z := newFloat(c)
// n goes up by two each loop.
for loop := newLoop(c.Config(), "atan", x, 4); ; {
term.Set(xN)
term.Quo(term, n.SetUint64(2*loop.i+1))
z.Add(z, term)
xN.Neg(xN)
if loop.done(z) {
break
}
// xN *= x², becoming x**(n+2).
xN.Mul(xN, xSquared)
}
return z
}
func (f BigFloat) String() string {
var mant big.Float
exp := f.Float.MantExp(&mant)
positive := 1
if exp < 0 {
positive = 0
exp = -exp
}
verb, prec := byte('g'), 12
format := conf.Format()
if format != "" {
v, p, ok := conf.FloatFormat()
if ok {
verb, prec = v, p
}
}
// Printing huge floats can be very slow using
// big.Float's native methods; see issue #11068.
// For example 1e5000000 takes a minute of CPU time just
// to print. The code below is instantaneous, by rescaling
// first. It is however less feature-complete.
// (Big ints are problematic too, but if you print 1e50000000
// as an integer you probably won't be surprised it's slow.)
if fastFloatPrint && exp > 10000 {
// We always use %g to print the fraction, and it will
// never have an exponent, but if the format is %E we
// need to use a capital E.
eChar := 'e'
if verb == 'E' || verb == 'G' {
eChar = 'E'
}
fexp := newF().SetInt64(int64(exp))
fexp.Mul(fexp, floatLog2)
fexp.Quo(fexp, floatLog10)
// We now have a floating-point base 10 exponent.
// Break into the integer part and the fractional part.
// The integer part is what we will show.
// The 10**(fractional part) will be multiplied back in.
iexp, _ := fexp.Int(nil)
fraction := fexp.Sub(fexp, newF().SetInt(iexp))
// Now compute 10**(fractional part).
// Fraction is in base 10. Move it to base e.
fraction.Mul(fraction, floatLog10)
scale := exponential(fraction)
if positive > 0 {
mant.Mul(&mant, scale)
} else {
mant.Quo(&mant, scale)
}
ten := newF().SetInt64(10)
i64exp := iexp.Int64()
// For numbers not too far from one, print without the E notation.
// Shouldn't happen (exp must be large to get here) but just
// in case, we keep this around.
if -4 <= i64exp && i64exp <= 11 {
if i64exp > 0 {
for i := 0; i < int(i64exp); i++ {
mant.Mul(&mant, ten)
}
} else {
for i := 0; i < int(-i64exp); i++ {
mant.Quo(&mant, ten)
}
}
return fmt.Sprintf("%g\n", &mant)
} else {
sign := ""
if mant.Sign() < 0 {
sign = "-"
mant.Neg(&mant)
}
// If it has a leading zero, rescale.
digits := mant.Text('g', prec)
for digits[0] == '0' {
mant.Mul(&mant, ten)
if positive > 0 {
i64exp--
} else {
i64exp++
}
digits = mant.Text('g', prec)
}
return fmt.Sprintf("%s%s%c%c%d", sign, digits, eChar, "-+"[positive], i64exp)
}
}
return f.Float.Text(verb, prec)
}
// Given a vector of real numbers x = [x_0, x_1, ..., x_n], this
// uses the PSLQ algorithm to find a list of integers
// [c_0, c_1, ..., c_n] such that
//
// |c_1 * x_1 + c_2 * x_2 + ... + c_n * x_n| < tolerance
//
// and such that max |c_k| < maxcoeff. If no such vector exists, Pslq
// returns one of the errors in this package depending on whether it
// has run out of iterations, precision or explored up to the
// maxcoeff. The tolerance defaults to 3/4 of the precision.
//
// This is a fairly direct translation of the pseudocode given by
// David Bailey, "The PSLQ Integer Relation Algorithm":
// http://www.cecm.sfu.ca/organics/papers/bailey/paper/html/node3.html
//
// If a result is returned, the first non-zero element will be positive
func (e *Pslq) Run(x []big.Float) ([]big.Int, error) {
n := len(x)
if n <= 1 {
return nil, ErrorBadArguments
}
// At too low precision, the algorithm becomes meaningless
if e.prec < 64 {
return nil, ErrorPrecisionTooLow
}
if e.verbose && int(e.prec)/max(2, int(n)) < 5 {
log.Printf("Warning: precision for PSLQ may be too low")
}
if e.verbose {
log.Printf("PSLQ using prec %d and tol %g", e.prec, e.tol)
}
if e.tol.Sign() == 0 {
return nil, ErrorToleranceRoundsToZero
}
// Temporary variables
tmp0 := new(big.Float).SetPrec(e.prec)
tmp1 := new(big.Float).SetPrec(e.prec)
bigTmp := new(big.Int)
// Convert to use 1-based indexing to allow us to be
// consistent with Bailey's indexing.
xNew := make([]big.Float, len(x)+1)
minx := new(big.Float).SetPrec(e.prec)
minxFirst := true
for i, xk := range x {
p := &xNew[i+1]
p.Set(&xk)
tmp0.Abs(p)
if minxFirst || tmp0.Cmp(minx) < 0 {
minxFirst = false
minx.Set(tmp0)
}
}
x = xNew
if debug {
printVector("x", x)
}
// Sanity check on magnitudes
if minx.Sign() == 0 {
return nil, ErrorZeroArguments
}
tmp1.SetInt64(128)
tmp0.Quo(&e.tol, tmp1)
if minx.Cmp(tmp0) < 0 { // minx < tol/128
return nil, ErrorArgumentTooSmall
}
tmp0.SetInt64(4)
tmp1.SetInt64(3)
tmp0.Quo(tmp0, tmp1)
var γ big.Float
e.Sqrt(tmp0, &γ) // sqrt(4<<prec)/3)
if debug {
fmt.Printf("γ = %f\n", &γ)
}
A := newBigIntMatrix(n+1, n+1)
B := newBigIntMatrix(n+1, n+1)
H := newMatrix(n+1, n+1)
// Initialization Step 1
//
// Set the n×n matrices A and B to the identity.
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
if i == j {
A[i][j].SetInt64(1)
B[i][j].SetInt64(1)
} else {
A[i][j].SetInt64(0)
B[i][j].SetInt64(0)
}
H[i][j].SetInt64(0)
}
}
if debug {
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Initialization Step 2
//
// For k := 1 to n
// compute s_k := sqrt( sum_j=k^n x_j^2 )
// endfor.
// Set t = 1/s1.
// For k := 1 to n:
// y_k := t * x_k
// s_k := t * s_k
// endfor.
s := make([]big.Float, n+1)
for i := 1; i <= n; i++ {
s[i].SetInt64(0)
}
for k := 1; k <= n; k++ {
var t big.Float
t.SetInt64(0)
for j := k; j <= n; j++ {
tmp0.Mul(&x[j], &x[j])
t.Add(&t, tmp0)
}
e.Sqrt(&t, &s[k])
}
if debug {
fmt.Println("Init Step 2")
printVector("s", s)
}
var t big.Float
t.Set(&s[1])
y := make([]big.Float, len(x))
copy(y, x)
for k := 1; k <= n; k++ {
// y[k] = (x[k] << prec) / t
y[k].Quo(&x[k], &t)
// s[k] = (s[k] << prec) / t
s[k].Quo(&s[k], &t)
}
if debug {
printVector("y", y)
printVector("s", s)
}
// Init Step 3
//
// Compute the n×(n−1) matrix H as follows:
// For i := 1 to n:
// for j := i + 1 to n − 1:
// set Hij := 0
// endfor
// if i ≤ n − 1 then set Hii := s_(i+1)/s_i
// for j := 1 to i−1:
// set Hij := −y_i * y_j / (s_j * s_(j+1))
// endfor
// endfor
for i := 1; i <= n; i++ {
for j := i + 1; j < n; j++ {
H[i][j].SetInt64(0)
}
if i <= n-1 {
if s[i].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
// H[i][i] = (s[i+1] << prec) / s[i]
H[i][i].Quo(&s[i+1], &s[i])
}
for j := 1; j < i; j++ {
var sjj1 big.Float
sjj1.Mul(&s[j], &s[j+1])
if debug {
fmt.Printf("sjj1 = %f\n", &sjj1)
}
if sjj1.Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
// H[i][j] = ((-y[i] * y[j]) << prec) / sjj1
tmp0.Mul(&y[i], &y[j])
tmp0.Neg(tmp0)
H[i][j].Quo(tmp0, &sjj1)
}
}
if debug {
fmt.Println("Init Step 3")
printMatrix("H", H)
}
// Init Step 4
//
// Perform full reduction on H, simultaneously updating y, A and B:
//
// For i := 2 to n:
// for j := i−1 to 1 step−1:
// t := nint(Hij/Hjj)
// y_j := y_j + t * y_i
// for k := 1 to j:
// Hik := Hik − t * Hjk
// endfor
// for k := 1 to n:
// Aik := Aik − t * Ajk
// Bkj := Bkj + t * Bki
// endfor
// endfor
// endfor
for i := 2; i <= n; i++ {
for j := i - 1; j > 0; j-- {
//t = floor(H[i][j]/H[j,j] + 0.5)
var t big.Int
var tFloat big.Float
if H[j][j].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
tmp0.Quo(&H[i][j], &H[j][j])
e.NearestInt(tmp0, &t)
tFloat.SetInt(&t).SetPrec(e.prec)
if debug {
fmt.Printf("H[i][j]=%f\n", &H[i][j])
fmt.Printf("H[j][j]=%f\n", &H[j][j])
fmt.Printf("tmp=%f\n", tmp0)
fmt.Printf("t=%d\n", &t)
}
// y[j] = y[j] + (t * y[i] >> prec)
tmp0.Mul(&y[i], &tFloat)
y[j].Add(&y[j], tmp0)
for k := 1; k <= j; k++ {
// H[i][k] = H[i][k] - (t * H[j][k] >> prec)
tmp0.Mul(&H[j][k], &tFloat)
H[i][k].Sub(&H[i][k], tmp0)
}
for k := 1; k <= n; k++ {
bigTmp.Mul(&t, &A[j][k])
A[i][k].Sub(&A[i][k], bigTmp)
bigTmp.Mul(&t, &B[k][i])
B[k][j].Add(&B[k][j], bigTmp)
}
}
}
if debug {
fmt.Println("Init Step 4")
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Main algorithm
var REP int
var norm big.Int
vec := make([]big.Int, n)
for REP = 0; REP < e.maxsteps; REP++ {
// Step 1
//
// Select m such that γ^i * |Hii| is maximal when i = m.
m := -1
var szmax big.Float
szmax.SetInt64(-1)
var γPower big.Float
γPower.Set(&γ)
for i := 1; i < n; i++ {
var absH big.Float
absH.Abs(&H[i][i])
var sz big.Float
sz.Mul(&γPower, &absH)
// sz := (g**i * abs(h)) >> (prec * (i - 1))
if sz.Cmp(&szmax) > 0 {
m = i
szmax.Set(&sz)
}
γPower.Mul(&γPower, &γ)
}
if debug {
fmt.Println("Step 1")
fmt.Printf("szmax=%f\n", &szmax)
fmt.Printf("m=%d\n", m)
}
// Step 2
//
// Exchange entries m and m+1 of y, corresponding rows
// of A and H, and corresponding columns of B.
y[m], y[m+1] = y[m+1], y[m]
for i := 1; i < n+1; i++ {
H[m][i], H[m+1][i] = H[m+1][i], H[m][i]
}
for i := 1; i < n+1; i++ {
A[m][i], A[m+1][i] = A[m+1][i], A[m][i]
}
for i := 1; i < n+1; i++ {
B[i][m], B[i][m+1] = B[i][m+1], B[i][m]
}
if debug {
fmt.Println("Step 2")
printVector("y", y)
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Step 3
//
// If m ≤ n−2 then update H as follows:
//
// t0 := sqrt( Hmm^2 + H(m,m+1)^2 )
// t1 := Hmm/t0
// t2 := H(m,m+1)/t0.
// for i := m to n:
// t3 := Him
// t4 := Hi,m+1
// Him := t1t3 +t2t4
// Hi,m+1 := −t2t3 +t1t4
// endfor.
if m <= n-2 {
tmp0.Mul(&H[m][m], &H[m][m])
tmp1.Mul(&H[m][m+1], &H[m][m+1])
tmp0.Add(tmp0, tmp1)
var t0 big.Float
e.Sqrt(tmp0, &t0)
// Precision probably exhausted
if t0.Sign() == 0 {
return nil, ErrorPrecisionExhausted
}
var t1, t2 big.Float
t1.Quo(&H[m][m], &t0)
t2.Quo(&H[m][m+1], &t0)
for i := m; i <= n; i++ {
var t3, t4 big.Float
t3.Set(&H[i][m])
t4.Set(&H[i][m+1])
// H[i][m] = (t1*t3 + t2*t4) >> prec
tmp0.Mul(&t1, &t3)
tmp1.Mul(&t2, &t4)
H[i][m].Add(tmp0, tmp1)
// H[i][m+1] = (-t2*t3 + t1*t4) >> prec
tmp0.Mul(&t2, &t3)
tmp1.Mul(&t1, &t4)
H[i][m+1].Sub(tmp1, tmp0)
}
}
if debug {
fmt.Println("Step 3")
printMatrix("H", H)
}
// Step 4
// Perform block reduction on H, simultaneously updating y, A and B:
//
// For i := m+1 to n:
// for j := min(i−1, m+1) to 1 step −1:
// t := nint(Hij/Hjj)
// yj := yj + t * yi
// for k := 1 to j:
// Hik := Hik − tHjk
// endfor
// for k := 1 to n:
// Aik := Aik −tAjk
// Bkj := Bkj +tBki
// endfor
// endfor
// endfor.
for i := m + 1; i <= n; i++ {
var t big.Int
var tFloat big.Float
for j := min(i-1, m+1); j > 0; j-- {
if H[j][j].Sign() == 0 {
// Precision probably exhausted
return nil, ErrorPrecisionExhausted
}
tmp0.Quo(&H[i][j], &H[j][j])
e.NearestInt(tmp0, &t)
tFloat.SetInt(&t).SetPrec(e.prec)
// y[j] = y[j] + ((t * y[i]) >> prec)
tmp0.Mul(&y[i], &tFloat)
y[j].Add(&y[j], tmp0)
for k := 1; k <= j; k++ {
// H[i][k] = H[i][k] - (t * H[j][k] >> prec)
tmp0.Mul(&H[j][k], &tFloat)
H[i][k].Sub(&H[i][k], tmp0)
}
for k := 1; k <= n; k++ {
bigTmp.Mul(&t, &A[j][k])
A[i][k].Sub(&A[i][k], bigTmp)
bigTmp.Mul(&t, &B[k][i])
B[k][j].Add(&B[k][j], bigTmp)
}
}
}
if debug {
fmt.Println("Step 4")
printBigIntMatrix("A", A)
printBigIntMatrix("B", B)
printMatrix("H", H)
}
// Step 6
//
// Termination test: If the largest entry of A exceeds
// the level of numeric precision used, then precision
// is exhausted. If the smallest entry of the y vector
// is less than the detection threshold, a relation
// has been detected and is given in the corresponding
// column of B.
//
// Until a relation is found, the error typically decreases
// slowly (e.g. a factor 1-10) with each step TODO: we could
// compare err from two successive iterations. If there is a
// large drop (several orders of magnitude), that indicates a
// "high quality" relation was detected. Reporting this to
// the user somehow might be useful.
maxAPrecision := 0
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
precision := A[i][j].BitLen()
if precision > maxAPrecision {
maxAPrecision = precision
}
}
}
if debug {
log.Printf("Max A precision = %d, precision = %d, tolerance %d, ratio = %.3f\n", maxAPrecision, e.prec, e.target, float64(maxAPrecision)/float64(e.target))
}
if float64(maxAPrecision)/float64(e.target) > 0.85 {
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
}
return nil, ErrorPrecisionExhausted
}
var best_err big.Float
best_err.Set(&e.maxcoeff_fp)
for i := 1; i <= n; i++ {
var err big.Float
err.Abs(&y[i])
// Maybe we are done?
if err.Cmp(&e.tol) < 0 {
// We are done if the coefficients are acceptable
var maxc big.Int
for j := 1; j <= n; j++ {
if debug {
fmt.Printf("vec[%d]=%d\n", j-1, &B[j][i])
}
t := B[j][i]
if debug {
fmt.Printf("vec[%d]=%d\n", j-1, t)
}
vec[j-1] = t
if t.Sign() < 0 {
t.Neg(&t)
}
if t.Cmp(&maxc) > 0 {
maxc.Set(&t)
}
}
if debug {
fmt.Printf("maxc = %d, maxcoeff = %d\n", maxc, e.maxcoeff)
}
if maxc.Cmp(&e.maxcoeff) < 0 {
if e.verbose {
log.Printf("FOUND relation at iter %d/%d, error: %g", REP, e.maxsteps, &err)
}
// Find sign of first non zero item
sign := 0
for i := range vec {
sign = vec[i].Sign()
if sign != 0 {
break
}
}
// Normalise vec making first non-zero argument positive
if sign < 0 {
for i := range vec {
vec[i].Neg(&vec[i])
}
}
return vec, nil
}
}
if err.Cmp(&best_err) < 0 {
best_err = err
}
}
// Step 5
//
// Norm bound: Compute M := 1/maxj |Hj|, where Hj
// denotes the j-th row of H.
//
// Then there can exist no relation vector whose
// Euclidean norm is less than M.
//
// Calculate a lower bound for the norm. We could do this
// more exactly (using the Euclidean norm) but there is probably
// no practical benefit.
var recnorm big.Float
recnorm.SetInt64(0)
for i := 1; i <= n; i++ {
for j := 1; j <= n; j++ {
tmp0.Abs(&H[i][j])
if tmp0.Cmp(&recnorm) > 0 {
recnorm.Set(tmp0)
}
}
}
norm.Set(&e.maxcoeff)
if recnorm.Sign() != 0 {
// norm = ((1 << (2 * prec)) / recnorm) >> prec
tmp0.Quo(&e.one, &recnorm)
tmp0.Int(&norm)
}
if e.verbose {
log.Printf("%2d/%2d: Error: %g Norm: %d", REP, e.maxsteps, &best_err, &norm)
}
if norm.Cmp(&e.maxcoeff) >= 0 {
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
log.Printf("Could not find an integer relation. Norm bound: %d", &norm)
}
return nil, ErrorNoRelationFound
}
}
if e.verbose {
log.Printf("CANCELLING after step %d/%d.", REP, e.maxsteps)
log.Printf("Could not find an integer relation. Norm bound: %d", &norm)
}
return nil, ErrorIterationsExceeded
}