// 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 (a *Mpint) SetFloat(b *Mpflt) int {
// avoid converting huge floating-point numbers to integers
// (2*Mpprec is large enough to permit all tests to pass)
if b.Val.MantExp(nil) > 2*Mpprec {
return -1
}
if _, acc := b.Val.Int(&a.Val); acc == big.Exact {
return 0
}
const delta = 16 // a reasonably small number of bits > 0
var t big.Float
t.SetPrec(Mpprec - delta)
// try rounding down a little
t.SetMode(big.ToZero)
t.Set(&b.Val)
if _, acc := t.Int(&a.Val); acc == big.Exact {
return 0
}
// try rounding up a little
t.SetMode(big.AwayFromZero)
t.Set(&b.Val)
if _, acc := t.Int(&a.Val); acc == big.Exact {
return 0
}
return -1
}
func mandelbrotFloat(a, b *big.Float) color.Color {
var x, y, nx, ny, x2, y2, f2, f4, r2, tmp big.Float
f2.SetInt64(2)
f4.SetInt64(4)
x.SetInt64(0)
y.SetInt64(0)
defer func() { recover() }()
for n := uint8(0); n < iterations; n++ {
// Not update x2 and y2
// because they are already updated in the previous loop
nx.Sub(&x2, &y2)
nx.Add(&nx, a)
tmp.Mul(&x, &y)
ny.Mul(&f2, &tmp)
ny.Add(&ny, b)
x.Set(&nx)
y.Set(&ny)
x2.Mul(&x, &x)
y2.Mul(&y, &y)
r2.Add(&x2, &y2)
if r2.Cmp(&f4) > 0 {
return color.Gray{255 - contrast*n}
}
}
return color.Black
}
// ToInt converts x to an Int value if x is representable as an Int.
// Otherwise it returns an Unknown.
func ToInt(x Value) Value {
switch x := x.(type) {
case int64Val, intVal:
return x
case ratVal:
if x.val.IsInt() {
return makeInt(x.val.Num())
}
case floatVal:
// avoid creation of huge integers
// (Existing tests require permitting exponents of at least 1024;
// allow any value that would also be permissible as a fraction.)
if smallRat(x.val) {
i := newInt()
if _, acc := x.val.Int(i); acc == big.Exact {
return makeInt(i)
}
// If we can get an integer by rounding up or down,
// assume x is not an integer because of rounding
// errors in prior computations.
const delta = 4 // a small number of bits > 0
var t big.Float
t.SetPrec(prec - delta)
// try rounding down a little
t.SetMode(big.ToZero)
t.Set(x.val)
if _, acc := t.Int(i); acc == big.Exact {
return makeInt(i)
}
// try rounding up a little
t.SetMode(big.AwayFromZero)
t.Set(x.val)
if _, acc := t.Int(i); acc == big.Exact {
return makeInt(i)
}
}
case complexVal:
if re := ToFloat(x); re.Kind() == Float {
return ToInt(re)
}
}
return unknownVal{}
}
// Evaluates a BBP term
//
// sum(k=0->inf)(1/base**k * (1/a*k + b))
func bbp(prec uint, base, a, b int64, result *big.Float) {
var term, power, aFp, bFp, _1, k, _base, oldresult big.Float
power.SetPrec(prec).SetInt64(1)
result.SetPrec(prec).SetInt64(0)
aFp.SetPrec(prec).SetInt64(a)
bFp.SetPrec(prec).SetInt64(b)
_1.SetPrec(prec).SetInt64(1)
k.SetPrec(prec).SetInt64(0)
_base.SetPrec(prec).SetInt64(base)
for {
oldresult.Set(result)
term.Mul(&aFp, &k)
term.Add(&term, &bFp)
term.Quo(&_1, &term)
term.Mul(&term, &power)
result.Add(result, &term)
if oldresult.Cmp(result) == 0 {
break
}
power.Quo(&power, &_base)
k.Add(&k, &_1)
}
}
// Returns acot(x) in result
func acot(prec uint, x int64, result *big.Float) {
var term, power, _x, _kp, x2, oldresult big.Float
_x.SetPrec(prec).SetInt64(x)
power.SetPrec(prec).SetInt64(1)
power.Quo(&power, &_x) // 1/x
x2.Mul(&_x, &_x)
result.SetPrec(prec).SetInt64(0)
positive := true
for k := int64(1); ; k += 2 {
oldresult.Set(result)
kp := k
if !positive {
kp = -k
}
positive = !positive
_kp.SetPrec(prec).SetInt64(kp)
term.Quo(&power, &_kp)
result.Add(result, &term)
if oldresult.Cmp(result) == 0 {
break
}
power.Quo(&power, &x2)
}
}
// 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())
}
// 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
}